ED051178.tif - ERICMaterial needed for an activity are listed with the activity description. Most...

DOCUMENT RESUME

ED 051 178 SP 007 241

TITL2INSTITUTIONPUB DATENOTE

EDRS PRICEDESCRIPTORS

ABSTRACT

K-12 Mathematics Curriculuw Guide.Reading Community Schools, Ohio.68210p.

EDRS Price MF$0.65 BC-$9.87*Curriculum Guides, *Elementary Grades, Grade 1,Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade7, Grade 8, Grade 9, Grade 10, Grade 11, Grade 12,Intermediate Grades, *Kindergarten, *Matnematics,*Secondary Grades

GRADES OR AGES: K-12. SUBJECT MATTER: Mathematics.ORGANIZATION AND PHYSICAL APPEARANCE: The guide is divided into threesections, one each for elementary grades, middle grades, and highschool. Each section is further subdivided by grade level. Sectionsare laid out in four columns across two pages. Column headings areconcepts, teaching methods and learning activities, resources, andevaluation. The vide is mimeographed and loose-leaf bound with asoft cover. OBJECTIVES AND ACTIVITIES: General objectives areoutlined in an introductory section. Suggested activities areco:related with specific mathematical concepts and specificobjectives. Most of the activities in middle grades and high schoolconsist of workia9 problems in textbooks. INSTEUCTIONAL MATERIALS:Material needed for an activity are listed with the activitydescription. Most materials listed for the middle grates ai.d highschool are page references in textbooks. STUDENT ASSESSMENT:Suggestions for evaluation accompany each group of activitiescorrelated with a concept--usually teacher observation in the lovergrades and teacher- developed and textbook quizzes in the uppergrades. (RT)

K - 12 WM S

Gurric.ilum Guide

1968-1969

S OEPARTMENTOF HEALTH,EDUCATION & WELFAIliOffICE Of EDUCATION

THIS DOCUMENT HAS REIN Pt,PRO,DUPE,' EXACTLY AS RECEIVED FILMTHE PERSON OP ORGONIZATiON DWGiNkTING IT POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILYREPRESENT OFFICIAL OFFICE OF FDLICATON POSITION OR POLICY

The Reading CoNit,nity Schools

Ile6d3mZ° Ohio45215

Robert G. Pickering, Superintendent

Ronald A. Hilvers, Dir:ctor of Instruction

Faculty Committee:

Robert Mixon

John Conley

Typists:

Doris Bennett,

Donald Filkins

Ronald Cutter

Bobby Sue Conley

William Lague

Nancy Krieger Janet Taylor

1

FORWARD

The pub'ication ofthis Curriculum Guide represents

the culmination of a year of study. lbe final editorial

work was carried out by six teams of teachers during

the summer of 163. Fhese materials were prepatad under

the supervision of Mr. Ronald A. Hilvers, Director of

Instructior.

Teachers are encouraged to refer ronseantly to this

guide during planning sessions throughout the year. It

should be understood that an:, resource material eannot

be the final ansi.::r to instructional questions, but

rather a tool to direct the thinking process of the

teacher. Periodically this guide will be updated. In

order to fazilitate this process teachers shouldvrite

in suggested changes, auditions, or d.qotions in the

spaces lea fot this purpotn. Particular attention should

be paid to sections of the guide t'ich outline expected

outcomes. By paying attention to these objectives, the

teacher should be better able to ensign and evaluate an

effective program for the children.

The Reading Community Schools

Reading, Ohio

September, 1968

2

PHILOSOPHY

MATH

The ptinary function of the modern mathematics

program is to develop in students the ability to apply

numbers and number operations intelligently and skill-

fully, and the ability to use quantitative procedures ef-

fectively in social situations both in and out of school.

The material in a modern mathematics program should

be structured to follow a logical sequence of presen-

tation, which leads to the most useful generalizations.

Pupils should be encouraged to develop the skill of

reaching e'eneralizations without teacher assistance.

Each pupil should develop ,o the fullest extent

of his mathematical capabilities. To achieve this the

individual needs of each pupil, whether college pre-

paratory or vocational, should be satisfied. These needs

will be net by offering as extensive and wide a variety

of mathematical exporienees as the pupil can successfully

complete.

Within these philosophical guidelines the Reading

Community Schools offer this Mathematics Curriculum guide.

-1-

3

EDUCATIONAL OBJECTIVES

1. Provide for independent study to promote originality,creativity, and thus revealing the uniqueness of theindividual learner.

2. Provide for the changing role of the teacher. This

involves a move from a dispenser of information roleto the role of a catalyst whose prime obligation isthe stimulation of the urge to inquire and the over-sight of individual independent st,.dy.

3. Recognize the worth of the individual student byenabling him to select the subject matter. This

will develop appreciation and a favorable attitudetoward the subject.

4. Provide for small. groups and thus teacher-pupilinteraction.

5. Begin a resource center accumulation taped lessons,recordn, word shr.ets, individual texts, etc.

6. Provide for the student and understanding of themathematically oriented, complex world in whichhe functions.

7. Provide the student with the mathematic skills toenable him to further his education or to functionas a productive individual in aocietv.

8. In addition to #6 and #7, to strive for the posses-sion in each student mathematic concepts, meaningand problem solving abilities.

9. Stress the analytical approach in the study ofmathematics.

4

Copceots of Arithmetic

Year

Readiness

Whole Numbers

Introduction to Fractions

Operations with Fractions

Introduction to Decimals

Operations with Decimals

The Integers

Modular Arithmetic

Different Number Bases

Commutative, Aaaociative, &

Distributive Properties

4 4.

.. V

I.4

..4.4

4o44

4.4

k.4

Place Value

Sots

Measurement

Percentage

Elementazy

Middle

8Secondary

XX

4IX

X...

.i--

X 4

+ -

X+

-X

+ -

X 4

+ -

X 4

+ -

X+

-X

+...

x 4

,

+ -

X

XX

XX

XX

..+

-X

'4

+ -

X 4

+ X+

-X

.+

-X

+ -

X.

-X

XX

X+

-X

+ -

X+

-X

4-

4--

X 4

-+

-X

4+

-X

4,X

4.

+ -

X 4

4.-

-X

4+

-X

4

XX

X

a, c

a, c

a,c,da,c,da.,coia,cid

a,c,d,a,c,da,c,d

a,c,d

a,c,d

a.c,d

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

X-

XX

XY.

XX

XX

XX

XX

XX

XX

_X

X

iX

XX

lX

Xx

FLW CHART

Concepts Of Geometry

Elementary

Middle

Year

12

34

_5

68

Recognition of Figures

Construction

Area

Volume

Inductive & Deductive Reasoning

Parallel & Perpendicular Lines

C")

Polygons

Circles & Arcs

Loci

Ratio & Proportion

Inequalities

Coordinate Geometry

Secondary

10

11

12

xx

xX

XX

XX

XX

, .,X

XX

XX

XX

XX

XX

XX

XX

X

XX

XX

XX

X-X

XX

X

XX

XX

XX

XX

X

XX

XX

XX

XX

XX

XX

XX

xx

XX

xx

x

xx

x

xx

xx

xx

x

xx

x

FLOW CHART

Concepts of Alsrebra

Elementary

Year

Symbols

Mathematical Sentences

Poly-m=1.31c

Special Products & Factoring

Algebraic Fractions

Graphing

The Real Number System

Function& Relation

Quadratic Equations

Logarithms

Progressions

Binomial Expansions

Matrices & Determinants

Permutations, Combinations &

Probability

Secondary

89

10

11

12

XX

XX

x,

xx

xx

xx

xx

xx

xx

xx

xx

xx

x

XX

XX

x

XX

XX

xx

xx

xX

xx

x

xx

x

xx

xx

xx

x

xx

x

XX

x

xX

X

xx

x

xx

x

FLOW CHART

Concepts or Advanced MathYear

Trigonometry

Functions of Angles

Graphs of Functions

Complex Numbers

Vectors

FUnft,me.:17,a1 Relations

Polar Coordinates

Solution of Trizngles

Analytic Geometry

Slope & Tangents

Equation of Straight Lines

Distance Between Two PrAnts

Conic Sections

Parametric Equations

Elementary

12

3

Mf.ddle

45

67

Secondary

89

10

11

12

x XX

x x XX

'

X xX

i

X x x X

FLOW CHART

Concepts of Advanced Math

Elementary

Middle

Secondary

Year

K1

26

78

Q10

11

12

C,lculus

Functions

Timits

Derivatives

Applications of Derivatives

Integratio:.

Applications of the Definite

Integral

Transcendentals

READING COMMUNITY SCHOOLS

CURRICULUM GUIDE

MATHEMATICS

ELEMENTARY

10

Concepts

Mathematics-KindergartenGeometry:Figures

- Investigating spheres, cubes,circles, squares, rectangles,and triangles- Investigating how figures arealike and how they are different

- Recognizing same shape: spheres,cubes, circles, squares- Investigating repeating patterns:shape

Time: approx. 5 days

Sets-Identifying the members of a set-Making sets of objects-Matching the membern of two setsto see if one has just as manymembers as the other-Making a set that has just as manyobjects as a given net

Time: approx. 3-4 days

Whole Numbers: 0 through 6- Investigating sets that have thesame number of members: zerothrough six

-Associating thc name with thenumber-Recognizing a pattern arrangementfor a number of objects

-Arranging objects in a patternto see if there are a givennumber of cbjects in a set

-Perceiving sets that: have thesame number


11

A) Teaching MethodsB) Learning Activities

A) Introduce assorted sizes and colorsof the various geometric figures.Permit children to manipulate,visualize and compare the dif-ference in sizes and shapes.

B) Through investigation the childrenfind that spheres and cubes donot look or feel the same.

A) Introduce the set by placingdifferent toys in three boxes.Each box should contain differenttoys and different numbers of toysExplain that each toy is a memberof the set; set is the collectionof the toys.

B) Use flannel board, yarn, and cutnuts to demonstrate how to makesets, comparison of sets and one-to-one matching of sets. The

children should be activelyinvolved in the manipulationof the objects and also to seea set with no objects.

A) Create a situation _n whichstudents have an opportunity toinvestigate seta that have thesame number of members. Theinvestigation should includevisual recognition, hearingpattern of sounds, feeling thepattern tapped on hand or shoulder.

B) 1. Clapping of hands. Each clapindicates a number 1=1 clap2. Display cutouts on flannelboard. Have children use yarn toenclose set.3. Have students place differentobjects in boxes or containersto demonstrate the idea ofmatching objects in any given set.

M e 1

A) PrintedB) Audio Visual

Resources C) PeopleD) Places

A) Greater Cleveland Math ProgramUnit 1 Teacher's Manual 1-16G.C.M.P., latest revisionpp. 1-10

B) Masking tapeBalls(assorted sizes)Blocks "

Spheres "

Rectangular BlocksWooden chalk boxesFlannel boardFlannel discs

A) Greater Cleveland Math ProgramUnit 2 Teacher's Manual 16-27G.C.M.P., Worksheetspp. 11-18

B) ToysBoxesWrapping paperMasking TapeYarnFlannel BoardFlannel Cut-outsTagboard (ut-outs

A) Greater Cleveland Math ProgramUnit 3 Teacher's Manual 27-43G.C.M.P., Worksheetspp. 19-30Key Topics in Mathematics forElementary Teachers SRA

B) Flrinnel boardFelt cut-outsYarnPicture cards showing 1,2,3,etc.ObjectsPins

Boxes or plastic containers

Several unrelated objectsLarge box

A) Expected OutcomeEvaluation B) Testing Program

A) Children demonstrate theability to recognize thedifferent geometric figures

B) Teacher Observation

A) Students exhibit the abilityto identify the member ,f aset, plaking of a set andmatching of members of two sets.

B) Anecdotal RecordsTeachers judgment

A) Children are able to recognizehow many objects are in a setfor sets having zero throughsix meseoers

B) Teacher judgmentObservation

12

M e 2

Concepts

Order Relation: 0 through 6-Matching members of sets todetermine which set has moremembers-Recognizing a set that has j'.'.ct

one more member than a given set.Arranging sets so that each hasone more member than the precedingset

-Investigating the natural orderof whole numbers through six

-Counting the objects in a set of six-Using numbers through six to identifythe position of an object in a row


Readiness for Addition-Separating a set of six or lessobjects into two parts-Identifying the number of a setand the number of each part of theSet

- Investigating sets of six orless objects to find ways toseparate them into two parts-Naming the number of objects ina set as a sum of the numbersof objects ic the parts



A) Have children match the objectsin two sets to decide which sethas just one more object thanthe other. Then they use thesedecisions to help them arrangesets in order so each set hasjust one more object than thepreceding set.

B) Play games to help children knowthe order of whole numbere1. Children form a circle bysitting around six large papersquares which have been glued tothe floor. On the 1st square puta star and explain that this ijsquare #1. Have students bouncethe ball on each square thenumber of times that will corerespond to that number of square.

Have children illustrate poemsthat relate to the order of thewhole numbers from 1-6For example see Teacher's Manualpp. 53

A) Provide opportunity for studentsto investigate the number ofobjects in a partitioned setand discover several sumsaesocidted with each number ofobjects

8) 1. Use flannel board and cutoutsto demonstrate how a sat of sixof 1-.ss objects nay be separatedinto two parts2. Display set of three blocksSeparate into two parts bystudents3. Hove students separate 0,PdSOn a string or wire into two DartsRefer to Teacher'll Manual .

pp. 54, 65

13

he3

2)



A) Greater Cleveland Math PrograwUnit 4 Teacher's Manual 43-53

G.C.M.P., Worksheetspp. 31-34Key Topics in Mathematics forElementary Teachers SRA

B) YarnFlannel. T:sard and Cutouts

Cards showing 0-6 objectsBeadEramePegboard and Pegs

A) Greater Cleveland Math ProgramUnit 5 Teacher's Manual 54-65G.C.M.P. Worksheetspp. 35-42

B) Flannel boardFlannel sut-outsBlocks9omino cardsHeavy wire rodWooden beadsCounting beads


A) Pupils demonstrate theability to do oral countingfrom 0-6 and to find the rightposition of a numeral

B) Teacher should observe and takenotes as she work' with thepupils

A) Students are able to separatea set of objects into two partsand to name the number of objectsin a set an a sum of the numberof objects e.g. Three is one plustwo

B) teacher observation and taking ofnotes on performance of students

14

M e 4

Concepts

Geometry: Shape and Size- Identifying figures that have thesame shape and size: circles, squares.

-Identifying the larger circle andthe larger square

- Testing for same shape: rectangles-Identifying the larger rectangleof two that have the same shape

- Testing for same shape: triangles-Identifying the larger triangleof two that have the same shape

_Arranging figures in order accordingto size: circles, squares


Whole Numbers: 7 through 12-Investigating sets that have thesame number of members: seventhrough twelve

-Associating the name with the number-Recognizing a pattern arrangementfor a number of objects-Arranging objects in a patternto see if there are a given numberof objects in a set-Perceiving sets that have the samenumber


Order Relation: 0 through 12-Matching members of sets to deter-mine which set has more members-Recognizing a set that has just onemore member than a given set-Arranging sets so thet each set hoeone more member than the precedingset

-Investigating the natural order ofwhole numbers through twelve-Counting the objects in a set oftwelve

- Using numbers through twelve toidentify the position of anobject in a row


15


A) Present the different geometricshapes. Guide children in theirinvestigation of the shapesane their compa ative size. Use

the flannel board and cut-outsto illustrate the desired concepts

B) Permit pupils to manipulate theconcrete objects to strengthentheir understandingRefer to Teacher's Manualpp. 65-72

A) Use counting objects for thenumbrals 7-12. Compare thenumber of objects in each set.Refer to Teacher's Manualpp. 72-89

B) Provide opportunities for pupilto make sets with counting obj_ctsand to compare their set withanother setUse rhythmic counting to music anddisplay the numeral that correspondswith the count.

A) Review the conceptthat 7 is 1 morethan 6, etc. for 7-12.Have children compare sets ofobjects with each set containing1 more then the previous setIntroduce the new words: eighth,ninth, tenth,

B) Cutout a tagboard doghouse andfix it so it will adhere to theflannel board. Place doghouse onflannel board and use yarn to maketwelve steps leading to doghouseMake 12 different colored tag-board puppies. Give one puppyto 12 children. Have the studentsplace the brown dog on the 1ststep, the red dog on the 3rd stepetc, thus,stressing the positionor order of the stepsRefer to Teacher's Manualpp. 89-96 M e 5



A) Greater Cleveland Math ProgramUnit 6 Teacher's Manual 65-72

G.C.M.P., Worksheetspp. 43-46

B) Concrete objects of circle, squarerectangles and triangles (Theseshould be of various sizes)

Flannel BoardFlannel cut-outs of rectangles,squares, circles, and triangles




A) Children demonstrate the abilityto identify the basic geometricshapes and their comparitive sizes

Children develop an awareness of therole of geometric figures ineveryday life

B) Visual testing of students

A)

B) Small counting objects as coloredcubes, dolls or other distinctivetoysFlannel board and numeral cut-outs B)

4by6 cards with the numerals 7-12


G.C.M.P., Worksheetspp.61-64

B) Flannel Board and cut-outsBead frame and beadsLarge objects to illustrate sets

1G

Pupils acquire the ability toassociate the name of a numberwith the numeralStudents are able to compare setsand distinguish between those thathave the same number of objectsand those which have less or more

ObservationAnecdote) Records

A) Pupils develop the ability towork with the sequence of numbersfrom 1-12

B) Teacher ObservationProgress made on worksheets ororal work

M e

Concepts

Numerals for Numbers: 0 through 6. -Recognizing the number of objects

in a set having 0 through 6-Associating a numeral with thenumber of objects in a set

-Choosing a numeral to tell how manyobjects in a set-Making a set of objects to show thenumber told by a given numeral-Arranging sets of objects andnumerals in order: 0 through 6


Another View of Addition-Separating a set of six or lessobjects into two parts

,Naming the. number of objects in aset as a sum of the numbers ofobjects in the parts-Selecting two sets to make a sethaving just as many members as agiven set

-Investigating the number of objectsin a part of a set when the numberin the set and the number in theother part are known

Time: apl.rox. 2-3 days

Numerals for Numbers: 7 through 12-Recognizing the number of objectsin a set haling 7-12 members

-Associating a numeral with the numberof objects in a aet-Choosing a numeral to tell how manyobjects are in a given aet: 0-12

-Making a set of objects to show thenumber told by a given numeral-Arranging sets of objects andnumerals in order: 0-12

-Investigating repeating patterns ofobjects that involve shape, size,and number


A) TeachinI MethodsB) Learning Activities

A) Introduce the numerals 0-6by placing felt numerals on theflannel board. Beside each numer31display a card showing the It ofobjects which correspond to thenumeral displayed. Say the. word

as the nmerals and picture girdsare displayed.

B) Provide many opportunities forthe children to see, hear andtouch the numerals 0-6

A) Display sets of objects which'children can partition into 2sets. Guide the children in aninvestigation to find severalways to exprese the number ofobjects in a set.Example: 3 is one plus two

3 is two plus one1 plus 2 is three

A) Provide in opportunity for thepupils to see, feel, say, and heareach mai:rat as is introduced.

Guide thou in their endeavor tofind a numeral for a given numberof objects. Use concrete objectsto help pupil gain knowledge of thenumerals 7-12 and their associationwith sets. Draw upon the pupilsprevious knowledge of concepts ofnumerals 0-6 to help broaden theirunderstanding

B)

17

Use rhythmic counting to music,clapping, tapping, cr jumpingrope to give meaning to thenumerals

M e l


Resoul:es C) PeopleD) Places



B) Flat.;Iel Board

Felt rumeralsFelt cut-outsYarnCards showing objects 0-6

Refer to Teacher's Manual pp. 96-110

A)

B)

Greater Cleveland Math ProgramUnit 10 Teacher's Manual 111-117

PegboardFlannel boardFelt cut-outsFelt numeralsConcrete objects(such as toys)katerials for making cards withnumerals to illustrate additivecombinations 2-6


G.C.M.P., Worksheetspp, i9-96

B) YarnFlannel BoardFlannel cut-outs & numeralsNumerals from sandpaperWooden blocks, beads, small toysCards showing 7-12 objects

18


A)The children are able torecognize the symbol for numbers0-6 and to use these numerals tocommunicate his ideas relatingto the numerals

B) Teacher observe the children'sability to recognize and use thenumbers 0-6Worksheets pp. 65-78

A) Pupils demonstrate the abilityto separate a set into two parts,choose numerals that tell thenumber of objects in the set andin each of its partsTeacher observes if the pupils

are able to express orally thenumber of objects in a set es thesum of the number of objects ineach part.

A) Pupils recognize and use thenumerals 7-12 and orally expresstheir orderStudents enjoy investigating atteripatterns of numbers

8) Appropriate dittosTeacher judgmentSkills performed on worksheetspp. 79-96

M e 8

Concepts

Mathematics-First YearReview- ore week

Pre-Number: One-to-One Matching-Recognizing and showing sets-Identifying members of a set-Using one-to-one corresponeence


Numbers and Numerals 1 through 5-Perceiving the numbers 1 through 5-Relating numerals to sets having1 through 5 members

-Recognizing and writing numeralsfor numbers 1 through 5

-Recognizing different numeralsfor 1 through 5-Ordering whole numbers 1 through5


Numbers and Numerals 1 through 7-Perceiving the numbers 1 through 7-Relating numerals to sets having1 through 7 members

-Recognizing and writing numeralsfor 1 through 7

- Recognizing different numeralsfor 1 through 7

-Ordering whole numbers 1 through 7


19


A) Demonstrate and teach differencebetween number and numeral

Teach names of numerals byrecognition

Demonstrate proper way of makingnumerals

Show meaning of each numeral byuse of sets learned before

B) Organizeigroups of childrencalling them sets of childrenShow complete set of children inroom, Set ca boys. Set of girls.

Children in one-to-onecorrespondence with desks in roomGroup children with a group ofbooks, pencils, etc.

A) See A) above

B) Use picture charts showingobjects of each number

Children can draw a set of 4, etc.Have children use cut-outs onflannel board to show a set of 5. .

Children use sticks, pireoleanersetc,to form sets at their seat

Give practice at board in formingnumbers then forming them on

paperDraw a picture illustratingeach set

Refer to activities above

M e 9


Pesources C) PeopleD) Places

A) rreater Cleveland Math ProgramUnit 1

G.C.M.P., Worksheets 1-4B) Use concrete objects within

classroom that are familiarto the children: themselves

deskbookspencils

A) Greater Cleveland liz,th ProgramUnit 2 pp. 5-23G.C.M.P.,Worksheets 5-24

B) Flannel boardCounting sticksCut-outsBoardPicture chartsRefer to text for othermaterials 5-23 Teacher's Manual

A) Refer to activities in Unit 2

20


A) Demonstrates ability to recognizea set of object, placing them inone to one correspondence

B) Observation by teacherSkills demonstrated on T.....)rk pages

in child's book

A) Exhibits ability to recognizenumerals and form sets for'numerals-Shows ability to formnumerals from memory

B) Observation of advancements madeSkills demonstrated in classroompractice and or work pages inbookAbility to use pnd recognizenumerals in classwork as observedby teacher

Me 10

Concepts

Numbers and Numerals 1 through 9-Perceiving the numbers 1 through 9- Relating numersls to sets having1 through 9 members

- Recognizing and writing numeralsfor 1 through 9

-Recognizing different numeralsfor 1 through 9-Ordering whole numbers 1 through9


Numbers and Numerals 1 through 12- Perceiving the numbers 1. through 12-Relating numerals to eets having1 through 12 members

- Recognizing and writing' numeralsfor 1 through 12-Recogrizing different numeralsfor 1 through 12-Ordering whole numbers 1 through 12


Geometry-The Number Linelthe segments

-Comparing len;;thu-Associating whole numbers withlengths-Perceiving the wenning of zero- Investigating order using thenumber line


21

A) Teaching MethodsB) Learning Activities-

_

Refer to activitiesiprevious page

Refer to aetivities,previoun page

A) Review different stapes

8) Draw incomplete shapes onboard having children completethe segmentDraw lines on board havingchildren judge longest andshortest, thus presenting sizeGive each child stri' . of construc-tion paper of assorted sires. Havechildren sort into groups of. onesize. Using these strips havechildren construct differentshapesExample: Make a rectangle. The

child rust find that hewill need 2 long stripsald two short stripsMake a square -thechild must find tat hemust have 4 strips of thesame size.

Moll




G.C.M.P., Worksheets pp. 37-49

A) Greater Cleveland Math ProgramUnit V Teacher's Manual 50-73

Tlorksheets pp. 51-70

A) Greater Cleveland Math ProgramUnit VI Teacher's Manual 73-91G.C.M.P., Worksheets pp. 71-90

B) Strips of colored paper(assorted sizes)ChalkboardRulers (number line)Contact paper (for number lineon floor)Magic Marker


A) Pupils show ability to recognizethe shapes of things around themPupils demonstrate their abilityto use the number line to help

them compare lengths and toinvestigate the.: order of numbers

B) Oral identification of the basicgeometric shapesTeacher observation and judgment

M e 12

A) Teaching MethodsConcepts B) Learning Activities

Ordinal Use of Whole Numbers-Using numbers to inoicateposition-Using the words first throughtwelfth

Time: approx.. 2 days

A) Review concept of numerals 1-12Establish orderof right to leftby 1.:,ing flannel board andflannel cut-outs. Assign countingnumber to each object. Show thatnumbers may he used to designateobjectsAsk questions such as: What isnumber 1? number 3?Change order of objects andcontinue with questions similarto that above

11) Draw large square on blackboard Mark off into columns withfive boxes in each column Havechildren number the rows fromleft to right.Have twelve students come to thefront of the room. Give onechild all cards with the numerals1-12. Have the child give each ofthe twelve children in front anumeral card- going from leftto right. Permit children totake turns doing the activity

Do work pages 91-94

Sum Through 5 A) Use flannel board and cut-outs to

-Investigating sums as numbers demonstrate the joining of two sets

-Investigating sums through 5 Display at top of flannel board-Investigating X as an unspecified numeral cards 0+1 through 11+1

number Choose or have a child choose thecard that corresponds to the set

Time: approx. 7 days which has been formed. Discussthe three numbers involved.Explainthat a set of three joined to a setof two forms a new set. The number3+2 is called a sum. Continue inlike manner to slic.w other sums.

B) ree the bead frrae to give practicein joining sots of differentcolored boa,1 utth MCP through SDraw five anises on the board. Putthe numerAl 5 in each chimney.Drawwindows in 0-5. Specifynumber needed to total five

Do worksheets pp. 95-108

M e 13


Resources C) People1)) Places

A) Greater Cleveland Math ProgramUnit VII Teacher's Manual 92-95

G.C.M.P., Uorksheets pp. 91-94

B) Flannel boardFlannel cut-outsNumeral cardsConcrete objects within theclassroon such as:Childrendesksbooks

pencilsBlackboard

A) Greater Cleveland Math ProgramUnit VIII Guide pp. 96-106


B) Flannel boardFlannel cut-outsBlocksToys

ChildrenConcrete objectsBead frameNumber lineNumeral cardsClothe' pins'

2'l


A) Demonstrate the readiness foruse of ordinal numbers.Shows ability to use the numbers1-12 to show position.

B) Observa(ion of the achievementon the behalf of the studentsas to the concepts taughtHow well the skills are demonstratedin classroom practice on board,worksheets or practice sheets

A) Students realize the joining oftwo sets rw-ult in a sum oftwo or more numbersChildren exhibit tt.e

to compute sums through fiveDemonstrate the ability !-ct findan unspecified number.

B) Teacher obsem.'ion of theskills mastered and abilityto use the skills on work.sheets or in class wrk.

M e

A) Teaching MethodsConcepts 13) Learning Activities

Subt':action A) Demonstrate aud teach the relation--Relating subtraction to addition ship of subtraction to addition-Relating sums and differences Example: 3+2=5through 5 5-3=2

5-2=3

Tice: approx. 4 days Introduce the concept of one-lessExplain that one le.s is theinverse or opposite of one moreDemonstrate the Pay to write theequatimsExplain the term difference

B) Have a group of children cone tothe front of room and join hands.Alternate joining 1 more to thegroup and one less child.Continue in like manner with sumsand differences through five.Children use counters plasticnumerals and cards at their deskto build sets and related sentences

Me1.5

25

A) PrintedB) Audio %,isual


A) Greater Cleveland Math ProgramUnit IX Teacher's Manual 106-115

G.C.M.P., Worksheets pp. 106-115Key Topics in Mathematics forPrimary Teachers

B) Flannel boardFlannel Cut-outsNumbar lineCountersPlastic numerals and symbolsWord cards with the word "is"on them

C


A) Students demonstrate the ability touse the relationship betweenaddition and subtractionPupils are able to complete simplemathematical sentences whichrelate the relationship betweenaddition and subtraction

B) Test ability of students to readand write addition and subtractionsentence by a simple teacher -madequizChildren test themselves on theability to recognize the one-moreand one-less patternstheir performance on the worksheetfor the teacher's evaluation aswell as their own

M e 16

Concepts

Plate Valle-Counting Tens-Counting tens to 12 tens-Using numerals for tens through12 tens (120)

-Computing sums aod differencesof tens through 5 tens-Investigating sums of tens and ones-Using standard numerals for sumsof tens and ones-Counting to 120

27


A) Revi w meaning of number andnumeralExplain the term computeIntroduce and demonstrate to theclass that it is possible tocount by tens as well ac ones.this should be done with concreteobjects such as sticks, papersticks, beads, etc.Introduce and show how to formthe numerals 10, 20, 30, - 120.

B) Have children count objects andplace one clothes pin for eachobject on the chart. When theyhave ten clothes pins exchangethe ten ones for one "ten" pinto be placed on the tens side(preferably another color of pin)Then children are to select thecorrect numeral card ( or maywrite the numeral on the board)which identifies the number ofpinsThis activity may vary and bundlesof sticks, paper strips and beadsetc. may be used instead of clothespins

Extend this kind of activity tohelp children understand countingby tens through twelve tens.Provide opportunity for studentsto practice writing numeralsintroduced.Have pupils arrange columns ofbeads ou the bead frame to showsums of multiples of tens.Send some students to the chalk-board to write the algorismafor columns of beads on bead frameas others manipulate the beads.Children compute the algorismson pp. 117-132

e



A) Greater Cleveland Math ProgramUnit 10, pp. 116-127

G.C.M.P., Worksheets pp. 117-132Key Topics in Mathematics forPrimary teachers

B) Pocket chartBundles of &ticksBead frameCounting menNumber kineClothes pinsChart for ten and ones made ofcardloaed

Numeral cards

281.


A) Children demonstrate competencein counting by tens throughtwelve tens or one hundred twentyStuden*,:s display the ability toform and use the numerals forsums of tens and ones

B) Teacher observation of skillsmanifested.Pupil perfoziance as indicatedby their work on worksheets orpractice sheets.

Men

Concepts

Sums of Six and Seven- Investigating sums for 6 and 7-Relating sums and differencesthrough 7- Investigating the order of addends-Investigating sums with 3 addends

2p

A) Teaching MethodsJ) learning Activities

A) We method described in Unit 8far investigation of sums for 6aid?Sae Unit 9-relating sums anddifferencesReview inverse relationshipconceptIntroduce commutative propertyof additionExample: 5+2=7 law oc

2+5-4 orderBegin by explaining that commutemeans to go back ar' Orth; there-fore the addends can hangs theirorder(or place) without changingthe sumExplain that more than two addendsare possible work example ofcolumn addition on the chalkboardUse concrete materials to reviewthe concept of joining of sets inrelation to addition of numbers

B) Have children take turns going tothe board to practice formationof numerals and formation ofnumerals aad the sums of six anasevenPermit children to use the variousteaching devices such a3: beadframe and beads, ten-frame, card-board cones, etc. to strengthentteir understanding of :findingstets

ME 19



A) Greater Cleveland Math ProgramUnit 11 Teacher's Manualpp. 128-144

G.C.M.P., Worksheets pp. 133-150Key Topics in Mathematics forElementary Teachers

B) Flannel boardFlannel cut-outTen framesCardboard cones.Bead frame


A) Students demonstrate the abilityto compute sums for six and seven.

13) Observation of advancements madeSkills demonstrated on work pays133-150

M e 20

Concepts

Sums of Eight and Nine-Investigating sums for 8 and 9-Relating sums and differencesthrough 9-Strengthening understanding oft:%e commutative property

-Strengthening understanding ofthe associative property

-Strengthening understanding ofthe closure property

-Strengthening understanding ofzero


Sums of Ten- Investigating sums for 10-Relating sums and differencesthrough 10

- Reviewing the meaning of 11through 19


31


A) Use methods described in Unit forinvestigation of sums for 8-9.

Demonstrate on chalkboard gt flannelboard the concepts of commutative,associative, closure, and identityproperty of zero. It is suggestedthat these be taught on separate

B) Give extensive practice for factsThrough sums of nine on boardand paperProvide opportun:Lty for childrento manipulate various concretematerials to help in theirinvestigation of sums of eightand nineGive oral practice of readingequations with the help of visualmaterials.

A) Use 10 objects. Have childrenarrange in 2 sets. Clearly showthat these arrangements of setsare another name for 10

Guide students in understandingthe commutative pronciple inwt".ting the addition factsReview the meaning of the numerals11-19 by oral discussion of thefact that 11 is 1 more than tenor (10+1), etc.

B) Give each child 10 objects such ascounters. Have th.. students

write the facts, then illustrate.Provide the opportunity for childto work with counters to observethe "undoing" or inverse relation-shipExample: 3+X=6 6-X=3

Use worksheets pp. 171-180 forpractice and display of theirCalls in computing suvs of 10

M e 21




G.C.M.P., Worksheets pp. 151-170Key Topics in Mathematics forElementary Teachers SRA

B) ChalkboardFlannel BoardFlannel cut-outs, symbolsand nomeralnBead frame and BeadsTen FrameCountingmen

A) Greater Cleveland Math ProgramUnit 13, Teacher's Manualpp. 164-172

G.C.M.P., Worksheets pp. 171-180Key Topics Mathematics forElementary Teachers

B) Large counting objectsSmall " "

Flannel BoardFlannel cut-outsNumber line

A) Expected OutcomeEvaluation /3) Testing Program

A) Students exhibit the ability tocompute ,mathematical equationsvertically and horizontally withthe sums of eight and nine.

Demonstrate through their work theunderstanding of the differentproperties taught in this unit

B) Teacher-made quizNow well the students use theskills on their worksheetsProgress of students As observed bythe teacher

A) Pupils show ability to compute theaddition combination through 10Students are able to tell andsolve simple mathematics sentence

Children display knowledge of theinverse relationship between aaddition and subtraction

B) Performance demonstrated ondaily worksheets and ditto sheetsleacher-made tests through seatworkand oral participation

Observation

M e 22


Sums Greater than 10-Investigating sums for 11 through20


A) Help children to visualize theprocess of addition withgreater than 10. Those who havedifficulty should have additionalopportunities to act out, '.isepictures or concrete objects togain a better ::nderstandingGroup class in order to meetspecific needs'Fork specific problems step iy Rtepon the board.

B) Drills through games such as"Number Ladder" and Fizz-BuzzBuild chart of factsWe counting men for helping thechildre3 visualize the concepts.Draw number line of floor. Permitchildren to explore one more, oneless concept

Money A) Exhibit the various coins and- Comparing the value of coins orally discuss the comparative- Using pennies, nickles, and value o1 the coin:dimes Guide the children carefully in

their recognization of pennies.

Time: approx. 1 day nickels, and dimes.

B) Set up a "play store" in the room.Children will enjoy bringingempty cartons, bottles, etc. tostock their store. Have childrenpractice buying objects with realor play money.Have children draw a line from eachcoin to the money word that matchesit. This may be done on the boardor at desks.

Me 23

33



A) Greater Cleveland Math ProgramUnit 14 pp. Teacher's Manualpp. 172-182


B) CountingmenChalkboaL.:..

White contact paper for number lineMagic MarkerConcrete objects which may bein classroom



B) Real money kit(ten pennies, 2 nickels, 1 dime)loy moneyEmpty cartons, bottles, cans etc,to stock storeCards with numerals labeled withthe cent sign(to use as tags forprice)

34


A) Pupils discover the addition factsfor sums of 11- 14. It shouldbe noted that the children are notexpected to master these factsat this point

B) ObservationInterest displayed by studentsTeacher judgment

A) Pupils are able to reco,penny, nickel, and dic,(

Children are able to cou,represented by coins ofdenominations such as [and nickels, or penni(to find amount of moneypay for an object

Students acquire the vLpertaining to the coin,

B) Teacher observationDaily worksheetsAbility to pay for obi(at splay store"

3

Concepts

Fractional Numbers-Meeting the counting problem-Recognizing the meaning of onehalf (1/2)

-Measuring lengths in inchesand half inches


Time

-Investigating the clock-Telling time to the hour andthe half hour

-Investigating the calendar (day,week, month)



A) Use the flannel board to illustratefractional parts. Guide pupilsin an oral discussion of howfractions apply in daily life suchas 1/2 of an ,pple or pie eatenpaper strips 1 1/2 inches longetc.

Explain fraction as a "part"

B) Pupils fold and cut paper to makefractional partsChildren divide objects on the boarC

Permit pupils to divide apples,bananas (just beforelunch)Practice oral conversationinv,Aving fractions

A) Use.flannel board and clock faceto illustrate how to tell time.

Give individualized help intelling time. For those who havegreat difficulty with this concept,it might be suggested that parentsgive help at night----6ne-to-onecorrespondence

Guide children through the investi-gation of the calendar

B) Children make their own clock facefrom a pie plate or cardboard.Guide children in the making ofa calendar.

35

M e 25



A) Greater Cleveland Math ProgramUnit 16 Teacher's Manualpp. 188-194G.C.M.P.,Worksheets pp. 203-206

B) Fractional d5scsFlannel board and cut-outsPaper, scissors, rulerConcrete objects such as bananaspies, apples, etc.Egg cartons which may be dividedinto halvesCounters to be divided into 1/2

A) Greater Cleveland Math ProgramUnit 17 pp. 195-200


B) Wall calendarsCardboard clock face withmovable handsTay clock with synchronized handsReal clockCardboard material to make acalendar

36


A) Pupils are able to find 1/2 ofan objectChildren mre able to apply theirknowledge of fractional partsto everyday livingBroaden their vocabulary toinclude ruler, inch, inches,measure, long, longer, longest, etc.Observation of work in classTeacher-judgment

A) Pupils are able to tell time to thehour and half- hour, record time ancdevelop an awareness of the passageof time-- hours through the daysChildrer. are able to associate tjmewith the evens of the dayAdd to their vocabulary o'clockhour hand, minute hand, noon,midnight, and half-past the hour.

13) Teacher observation of how accuratethe pupils can tell time bothorally and in written form

Me 26

Concepts

Sums of Eleven- through Twelve-Using sums for 10 to investigatesums for 11 and 12-Using the associative property-Relating sums and differences for11 and 12


A) Teaching MethodB) Learning Activities

A) Use the flannel board or chalk-to illustrate how a set of 8 membersand a set of 7 members are joined.Help the children to see that re-arranging a set so that they cansee a group of ten within helps themto add.

Place three sets in the flannelboard-Ask children:]. How many setsof ten can you find? come & circle

2. What is anothename for one ten and five ones?

3. Write a numbersentence to show our thinkingReview associative property ofaddition, also combinations offacts whose sums are 10.

B) 1)ovida many opportunities for .

students to investigate problemswhcse sums are greater than 10Use worksheets pp. 211-226 to providepractice EV': children

Use chalkboard to give opportunityfor pupils write math sentencesand to demonstrate their skill inusing the associative property tohelp solve simple equations.

37

N e CT



A) Greater Cleveland Math Programpp. 201-210


B) Flannel board & cut-outsConcrete objects sich as: beadsclother pins, buttons, sticks,strips of paper, etc.ChalkboardCountingmenNrmber lineMaterial for making cards toshow equations such as:

44.1=X or (5+5)+3=13


A) Pupils are able to compute simpleequations whose sums are 11 and 12Children are able to use theassociative property to help themfind sums greater than 10

Students us:, the knowledge of ournumeration system as based on tento help them find sums greater than 10

B) Children test themselves on theirability to compute problems whosesums are 11 & 12. They present their

work to the teacher for evaluationof their performance

M e 28

Concepts

Sums of Thirteen and Fourteen-Using sums for 10 to investigatesums for 13 and (4-Using the associative property-Relacing suos and differencesfor 13 and 14


Sums of Fifteen through Eighteen-Using sums for 10 to investigatesums for 15 through 18-Using the associative property-Exploring patterns in additioncombinations-Computing sums of 3 addends


39

A) Tea,:hing Methods

B) Learning Activities

A) Start with sets of concreteobjects. Htlp the childrenregroup these sets and relatethe appropriate mathematicalsentence to each step of hiswork. Careful guidance shouldbe used to help the pupils learnto use the thinking steps involvedin adding two numbers su,:h as 9and 4Help the students discover theinteresting patterns by helpingthem to find that for a givensum, as the first addend becomesless by one, the second .kddendbecomes greater by one.

3) Use worksheets 227-246 to helpstudents find patternsPlay the game I'm Thinking of aNlmber to help the children thinkof numbers.Use number line to help studentsadd and subtract sums greaterthan ten. ( Each child maybenefit by his own individualnumber line- ruler may be used)

A) Use same method as described forprevious Units

For other suggested teachingmethods see G.C.M.P. guide pages222-235

B) Use wordsheets pp.247-266Give students the opportunity tomanipulate various concreteobjects such as the Ten Frameto help in the developmentof the hrder combinations

M e 29



A) Greater Cleveland Math ProgramUnit 19


B) CountingmenNumber lineHundred boardCounting objects such as milkbottles caps, buttons andcolored sticks

RulersFlash cards and materials forchildren to make their cwn.Flannel board and cut-outs

A) Greater Cleveland Math ProgramUnit 20 pp. 222-235

G.C.M.P., Worksheets pp. 247-266Key Topics of Mathematics forElementary Teachers

B) Pocket chartTegboard equation cars(for each child)

Ten FrameNumber lineLarge pegboardColored spools

40


A) Pupils demonstrate the ability touse sums of ten to help themfind the sums for thirteenand fourteen

Students use their knowledge ofthe associative property ofaddition to help them tind sumsgreater than-10

Children exhibit an understandingof the inverse relationship ofaddition and subtraction

B) Skills performed on workpages227-246

Teacher-make quizObservation by teacher on class-room work and the ability to

use these skills and previoustaught ones

A) The cl,ildren are able to apply

their understanding' of -the'

associative, commutative, andone-more, one-less patternto help them work with combination

that result in sows of 15 or 16Pupil develops an ability torecognize the inverse relationshipbetween addition and subtraction

Pupils are able to compute sumsof three addends

B) Teacher observationSkills performed on 1.orkshcets,dittos, or board work

Keep individual case records

14 e 30

Concepts

Skip Counting-Counting by tens-Counting by twos-Counting by fives


41

A) Teaaing MethodsB) Learning Activities

A) Review the number sequence whichbegins wiL.: 10 and increases by10 each time or decreases by 10Review one-by-me counting and theorder of the numbers. Help thepupils discover the pattern ineach sequence. Use the same methodto help with patterns of 10's,2's and S'sProvide bundles of sticks, each tocontain 10 sticks to give childrenpractice in counting by 13's'71o:instruct a hundred chart in 24 in.

by 3ti in tagboard. Discu3s theusage of the chart in her;ing thestudents find the number sequenceor patternsDraw ladders and steps in the boardHave the students find tha missingtert -3 and write the correvondingnumerals on the ateps or Ladders.(This is best for children towork in small groups or tiled as

relay games)

e 31



A) Greater Cleveland Math P.)gramUnit 21 Guide pp. 236- 240


B) Bundles of sticksSheet of tagboard (24" by 36")Flannel Board and cut-outsConcrete objectsCardsMultiples of 5-100

Counters

A) Sxpected OutcomeEvaluation B) Testing Program

A) The pupils are able to skip countby 2's, S's and 10'sThe children exhibit the abilityto recognize the number patternsboth increasing and decreasing

B) Oral testingWorksheets

42

Me 32

Concepts

Mathematics Second YearI. Addition of Whole Numbers

-Union of sets to define addition-Sums of ten, nine, eight, seven,

and six-Inverse relationship (Informal

presentation)-Missing addend problems-Column addition- one digit-

four numerals-Computing sums and differences

0-10

Time: approx. 2 1/2 weeks

II. Zeros in Addition-Using number line to count by

7's

-Using a calendar-Two-digit addition- no carryingusing zero as a place holder

Time: approx. 1 week


A) Use flannel board or chalkboard todemonstrate the joining of setas a means of finding the sums

Introduce, informally the inverserelationship 2 and 1 is 3

3 less 1 is 2Make into number sentence:

2 and 1 is

3 less 1 is

Work as many examples on the boardas necessary for students tograsp the concepts. Have studentsgo to board and work problems.Observe carefully those who .eedextra help.

Explain how to use a number line bytelling students to move to theright for addition to the left forsubtraction

B) Use worksheets S1-32 for practicePermit students to manipulateconcrete materials to aid inaddition

A) Illustrate with counting sticks inbund; of ten.

Use counting men to demonstrate placevalue or may vse glace value chartsDemonstrate using the number lineto add multiples of ten.

Review empty setExplain 30 as no ones and adding of

+40

3 and 4 to find the number of tens

B) Clasa take turns working examplesat the board.Teacher observationUse worksheets S 33- 42 Eor practiceMake ditto sheets Eor supplementaryworksheets

43

Me 33



A) Greater Cleveland Math ProgramSecond Year, Latest Edition

G.C.M.P., Worksheets pp. 51-32Key Topics in Mathematics forFrimary Teachers, SRA

B) Number lineCounting objectsFlannel boardFeit numerals and cut-outsCounting sticksAbacus

A) Greater Cleveland Math ProgramG.C.M.P.,Worksheets pp. S33-42Key Topics in Mathematics forPrimary Teachers, SRA

B) Counting objectsAbacusNumber lineCountingmenFlannel board & array cut-outsCounting sticks

single sticksbundles of sticks

4


A) Students are able to cowpute sumsof 0-10

Students exhibit an understandingof the inverse relationshipbetween addition and subtractionand are capable of pr:s.dicting the

outcome of a number sentence

B) Teacher Oservation of work ofthe board

Performance on worksheetsTeacher-made quiz

A) Pupils demonstrate the ability tocompute sums greater than lo-using zero in ones place

B) Teacher observationApplicLtion of gained know/edgeto new situationsTeacher-made quiz

Me 34

Concepts

III. Addition and SubtractionSums and Differences 10-18

-Using sets to find sums anddifferences

-Using number line to completemathematical sentences

-Computing of sums and differences-Story excerises involving

addition and subtraction-The addition algorism is develop-

ed through the use of arraymodels or sets.

-The subtraction algorism isdeveloped through the use ofmodels

Time: approx. 2-3 weeks

45


A) Illustrate examples of problems onthe board. Use bundles of sticksand place value charts to helpchildren visualize the thinkingProcess.Let children pretend that erasersor other objects are numeralsLet them dramatize number problems

Review the concept of inverserelationshipUse the number line to illustrateproblems5 +7= (545) +2 as a thinking process

thus the informal introduction ofthe associative propertyExplain how the associative propertywill help on column. addition sinceaddition is a binary process.

B) Worksheets 1,3. 43-94

M e 35



A) Greater Cleveland Math ProgramG.C.M.P., Worksheets pp. S43-94Key Topics in Mithematics for

Primary Teac,.,rs, SRA

B) Flannel boardFelt cut-outsPlace value chartCountingmenNumber lineAbacusBundles of sticksCounting beads

Filmstrip"Two Properties of Addition"

46


A) Students exhibit the abilitycompute sums and differencesof 10-18

B) Daily observation by the teacherTeacher judgmentTeacher TestApplication of skills on worksheets

He 36

Concepts

IV. Two-Digit Addition With Carryingto Tens and Hundreds Place

-Using models to develop theaddition algorismArray modelsNumber strip


47


A) Have children express essence ofthe problem.34 34 and 52 is how many?

+52 34 and 52 is

Emphasize the placement of thesign in the example and the orderlyarrangement of the two numerals.Go from the above type example toone like this:38=30+8

+54=50+480+12 (80+10)+2=

Explain carefully the regroupingprocess. Work as many examples asnecessary for children to get theconcept. Have the children go tothe board, work examples & explainthe thinking process:Teacher should observe the studentscarefully to give additional helpwhen needed.Use array cut-out model to developthe addition algorism-or may useumber strip model.Use counting men to stress placevalue and sums of tens and ones.

B) Provide opportunities for studentsto take turns placing the fingerson the countingmen while anotherstudent writes the equation on theboard-a third child writes expandednotation and a fourth childcomputes the sumPrepare ditto sheets for additionalpractice to pp. 43-94. Give muchoral as well as written practicefor this.

M e 37



A) Greater Cleveland Math ProgramG.C.M.P., Worksheets pp. 95-104Key Topics in Mathematics for

Primary Teachers, SRA

B) Counting menPlace value chartsCounting sticksConcrete objectsNumber lineAbacusFlannel board and cut-outs

48


A) Pupils display Pn understandingof the principles involved inregrouping from ones to tenstens to hundreds

Students are able to compute 2digit addition with carrying,successfully

B) Self-evaluation Ly the studentsTeacher take notes on studentsability to apply their under-

standing of the principlesinvolved in regrouping to

compute problems on worksheetsTeacher Tests

24 e 38

Concepts

V. Measurement-Money

Computing sums with money-Linear measurement

Using 1/2" to measure lengthUsing 1" to measure lengthObserving that whole numberscan't be used to answer allthe questions about how many orwhat length


A) Use real or play money to helpstudents visualize what amount ofmoney they are to computeDiscuss money problems orally inclass. Work prolAeme step by stepUse yardstick or rulers as acomputing device to measure thingssuch as their books, pencil, etc.

Compare lengths of. edges of squares,triangles and rectangles for a

-GeometryComparingsquares,

-Recognizingobject

lengths of esges oftriangles and rectangles1/4 and 1/2 of at.

class demonstration.Review fractional measures of 1/4

and 1/2. Use discs for visualclues


VI. Addition of Whole Numbers-Computing sums with two, three,

and four addends.-Reconstruction Problems-Correcting computations of

pre-worked problems-Ordinal Numbers


49

B) Have students work pp. 105-116Use geometric figures for art class.

A) Discuss problems orally in class.Work problems step by step. Guidethe students through the thinkingprocesses of addition. Call to theattention of the students thataddition is a binary process.

Use array modeis or cthalkboard toillustrate the steps in working the

problems.23 ADD ALL ONES FIRST AND33 PUT THE WHOLE ANSAR DOWN

+12

8 ADD ALL TENS AND PUT DOWN60 WHOLE ANSWER (may be68 reversed)

B) Provide many opportunities for boardwork practice sheets and oral drillon filets.

e



A) Greater Cleveland Math ProgramG.C.M,F.,Worksheets pp. 105-116Key Topics in Mataematics for


B) Money kieeither real or play)RulersNumber lineFlannel board & cut-outsGeometric figures- rectangle,

triangle, and square- (thesemay be concrete or semi-ccacrete)

Fractional discsBooks, pencils, etc. formeasuring

A) Greater Cleveland Math ProgramG.C.M.P., Worksheets pp. 117-132Key Topics in Mathematics forPrimary Teachers, SRA

B) Flannel boardArray cut-outsChalkboardAbacusFlash cards


A) Pupils are able to make comparativemeasures in a meaningful wayand to apply this knowledgeto every day situations.

Students demonstrate the abilityto compute sums involving money,through direct and vicariousexperience

B) Weekly tests (teacher made)Diagnostic testsSeli-evaluationObservation by teacher of pupils

work

A) Students become stcongly motivatedtoward learning arithmeticfacts, skills and processes.

Pupils are able to grasp theconcept of addition as f, binaryprocess and apply this tocolumn addition

B) Direct observation of the childrenas they are engaged in thelearning activity is the bestappraisal method

Weekly teacher-made testWorksheets and ditto sheets

M e 40

Concepts

VII. Measurement-Time;

Investigating intervals of timeon a clockMinutes before the hourMinutes after the hour


VIII. Subtraction of Whole Numbers- Two -digit subtracting without

borrowing-Place value of ones, tens, and

hundreds-Using pennies and dimes to

emphasize ones and tens-Story problems-Recognizing differences as missing

addends



A) Discuss hour hand and minute hand.Use actual happenings to show timeExplain carefully the intervalsof time such as 1/2 hour, 5 minutesetc.

Demonstrate how to write timeExample 12:30 or 12:05Give studentsan opportunityto practice writing aswIll as

telling time.

B) Let children make own clock and

show timeDo worksheets pp. 133-137

A) Review concept of difference.Use chalkboard to demonstrateproblems step by step

Number line may be used toexplain differenceExample 30-20 It is the number

when added to 20 will give 30.Show on flannel board sets ofdifferent colored discs-only onecolor show at a time. Compare

sets to discover "how many more?"Use number line and chalkboard tocarefully develop the subtraction

algorism involving borrowing.Stress the fact that when centsis subtracted from cents the answer

is always cents

B) Have students go to board 3-6 at a

time. Give every other student

the same problem. Have otherstudents work at their desks.Teacher should dictate problemsto solve.Carefully observe the studentsand regroup class for meetingspecific needs of students.

e 41



A) Greater Cleveland Math ProgramG.C.M.P., Worksheets pp. 133-137Key Topics in Mathematics for


B) Large clock with movable handsSmall individual clocksPaper to make clock facesPaper fasteners for clock hands

A) Greater Cleveland Math ProgramG.C.M.P., Worksheets pp. 137-156Key Topicsin Mathematics for


B) Place value chartChart with questions written on

it to help students with storyproblems

Number lineCounting sticksCards showing sets of coinsFlannel board, felt numbers

with cent symbols

Filmstrip"Place Value and Subtraction"


A) Pupils can distinguish betweenthe hour and the minute hand

Pupils can tell time to the hourhalf hour, and five minuteintervals

8) Test by making clocks and havehands drawn in to designate

timeOral tests

A) Pupils demonstrate competencein finding differences

B) Self-evaluation (children gradetheir own papers and correctmistakes

Teacher made tests weeklyObservation of work on board

and evaluation of daily papers

1.1 e 42

Concepts

IX. Addition and Subtraction-InNestiEation a missing addend

as a difference and differencesas a missing addend (inverserelationship)

-Using sums and differences inapplication situations


oorn


A) Use chalkboard to demonstrate theinverse relationship between ad-dition and subtraction.

Class discussion and working ofproblems on blackboard.Explain equation form 16 - 8 = 8(equation)

B) Have pupils work pp. 156-166 forpractice.Use flannel board and discs tovisualize computational procedurein computing sums and differences.Money kit may also be used.

Have pupils take turns going to thechalkboard, flannel board orcountingmen to demonstrate theirunderstanding of computing sumsand differences and the inverserelationship of addition andsubtraction.

MoL8



A) Greater Cleveland Math ProgramGuide pp. 156.164G.C.M.P. worksheets 157-166Kev Tonics in Mathematics forPrimary Teachers, SRA

B) CountingmenFlannel boardTagboart. discs colored copper andsilver to simulate pennies anddimes (ones and tens)Number line on board or floorCards labeled tens and onesMoney kit (either real or playmoney)

J4


A) Pupils are able to use theirknowledge of the inverse relationof addition to subtraction toimprove their computationalaccuracy.

B) Teacher judgment by observationPerformance on worksheets pp.157-166Self evaluationTeacher made quiz

Me 44

Concepts

X. Computer Mathematics-Exploring the use of a computer

made with two rulers-Using rulers to compute sums

and differences


55


A) Draw two large rulers on thechalkboard to illustrate thecomputing device. Draw one rulerat a time. Have students place thetheir 12" rulers in the sameposition as the ones you have drawnon the board. Call attention tozero as the starting point.Carefully demonstrate and explainthat by moving the lower rulerto the right represents additionExample: Slide the lower ruler oneinch to the right so that the 0on thelower ruler falls below the1 on the top ruler. This willcompute all the sums 1+2=3, etc.When the students are ready you mayshow how to find 2+ 4=6 by slidingthe lower ruler 2" to the rightand below the 0 on the top rulerContinue in like manner by sums of3+; 4+-, etc.

lInview the inverse relationship ofaddition and subtraction.Demonstrate how the pupils may usemechanical computation ofdifferences such as 7-6=1 withoutchanging the position of the rulers

8) Let children work in groups of 2 or3 to measure their paper rectanglesand to discover how to find thecombined lengths by using acomputing device. Give much practic

Workpages 167-178

14e45



A) Greater Cleveland Math ProgramG.C.M.P.,worksheets pp. 167-178

B) Paper rectangles (3" by 4" and5" by 6")

2 rulers 12" long (for each child)2 yard sticks

56


A) Students are able to use theirknowledge of counting numbersto check what he discovers onthe computer. (Finding thecorrect answers is not ofmajor impirtance in this unit)

Pupils become familiar withmechanical computation

Visual demonstration of pupilsability to operate a mechanicalcomputer

B) Teacher observation of interestskill3 and familiarity ofprinciples involved in mechanicalcomputation

He 46


XI. Place Value-Counting Hundreds-Counting hundreds and counting

tens through 1800-Using numerals for hundreds-Computing sums and differences

of hundreds-Recognizing numbers as sums of

hundreds, tens and ones-Writing standard numerals for

sums of hundreds, tens, andones

-Associating numbers with distancesfrom 0 on the number line


A) Introduce conventional names forcounts of hundreds on chalkboard.Explain to students that additionand subtraction combinationsapply to hundreds as well as onesand tens.

Demonstrate on chalkboard that whenhundreds are added to hundredsthe sun. will be hundredsUse counting men or place valuechart and cards to illustrateplace valueHave children take turns going tothe board to write the numbersthat the teacher dictate: whileanother student places thecorrect fingers on the countingmen or uses the place value chartand number cards.

B) Have students work pp.179-192for practice

Play matching game, Write a

column of standard numerals onone side and to the right ofthis column write the expandedformTake turns calling on children toread the standard numeral and tomatch that numeral with theexpanded form

Play relay race with members ofeach team writing expanded formof numbers teacher dictates.

14 e 47

A) PrintedB) Auaio Visual


A) Greater Cleveland Math ProgramG.C.M.P.,Worksheets pp. 179-192

B) Counticog menNumber lineFlannel board and felt numerals100 sticks or counting objectsPlace value chartNumeral cardsBead frame and beads

Filmstrip"Place Value and Subtraction"

A) Expected OutcomeEvaluation B) Testirg Program

A) Pupils display the ability to ccount by hundreds, write numeralsand compute sums and differences

B) ObservationTeacher-made testsWorksheets pp. 179-192Self-evaluation by student

58

M e 48

ConceptS

XII. Addition-Investigating sues of hundreds,

tens, and ones-Computing 3ums of hundreds, tens,

and ones-Investigating addition algorisms

Time: approx. 6-7 day.

XIII. Practice Work Linear Measurement-Measuring lengths in inches, feet

or yards-Computing sums of lengths and

differences of lengths



A) Demonstrate on the chalkboard howto compute three-digit addends.Begin by showing the tht.kingprocess and using the lo q algorism;156 Then shorten: 150

+387 +387

13

130400543

543

B) Students work pp. 193-206After pupils have finished theirwork they should have their workchecked by the teachers manualor answer sheet, circle incorrectanswers, and then correct theirmistakesIt is suggested that the teacherdiscuss each new concept withchildren and go over each problemwith the students carefully sothat a good foundation may belaid in working addition involvingregrouping or carrying.

A) Explain by means of large-sizedruler on the board.

Demonstrate how to measure objectsby using the ruler or yardsticks

B) Provide many and various oppor-tunities for pupils to measureobjects in the room such as:desks, bookcase, books, etc.

K e 49



A) Greater Cleveland Math ProgramG.C.M.P., Worksheets pp. 193-206

B) Pocket char;.Hundred-cards and ten-cardsFlannel boardFelt discsTagbcard set

union outline

A) Greater Cleveveland Math ProgramG.C.M.P., Worksheets pp. 207-214

B) 12" rulersYard stickChart of measure(made by teacher)

A) Expected Outcomevaluation B) Testing Program

A) Pupils become proficient in theuse of the addition algorismwith three-digit addends

B) Teacher-made quiz to test theability of students to computesums of hundreds

Teacher observation of pupilsday-to-day improvement in hiswork whether it is boardwork, class discussion orworksheets pp. 193-206

A) Pupils develop an awareness ofmeasurement and can apply thisknowledge of measurement tocomputation of lengths

Children improve computationalskills through practice

B) Visual & oral demonstrationsPerformance on worksheetsTeacher observation

60

M e 50

Concepts

X1V. Ordinals and Time-Using numbers to show order and

position-Using numbers to investigate

order and position of daysin a year

-interpreting time shown on aclock

Time: approx. 4-5 cloys

XV. Fractional Number-Recognizing thirds and halves-Using inches and third-inches

to measdre lengths


c1


A) Explain ordinal numerals informallycounting - 1. 30

Ordinal - 3.st 30th

Provide various opportunities forstudents to use ordinal numbers

and numerals to show position.Draw large grids on the board.Number the 2irst three grids.Have students help complete thegrids. Let this experience leadto the making of calendars.Discuss days of the months& year

Use large clock to review conceptof time

B) Have pupils make clocks withmovable hands to set at varioustimes

Play games with students involvingtime-see Manual pp. 209-210

Call attention to times to the daywhen special events take placeso students can see applicationof time to Rveryday livingPupils work pp. 215-224,practice

A) Explain meaning of term "fraction"to mean a"part of"

Draw circles, squares, & rectangleson the board to show equal partsof 1/3, 1/2 and their relationshipto a whole.

Explain carefully what the 1 meansand what the 2-3 means in 1/2, 1/3

EmphaP4ze the word "equal" whenreferring to 1/3 of a whole as 1 of3 parts which are equal

S) Fold paper into thirds and halvesDivide sets into thirds and halvesDivide groups of students intothirds and halves.

Exercises from chalkboardWorksheets pp.225-230

He 51

A) PrintedB) Audio VIsual


A) Greater Cleveland Math Program

Teacher's Manual pp. 203-210G.C.M.P. worksheets pp. 215-224

Ka T22412inJathematiaa_EaPrimary Teachers, SRA: order andposition

B) CalendarsClocks with movable hands

A) Greater Cleveland Math ProgramTeacher's Manual pp. 211-217

G.C.M.P. worksheets pp. 225-230B) Fraction chart

Concrete objects which car, bedivided into equal parts

62


A) Children broaden their understand-ing of ordinal numbers by workingwith calendars and clocks.

B) Test by making clocks and havehands drawn in to designate time.

Oral tests on time and ordinalnumbers.

Teacher obcorvOlon of daily work.

A) Students are able to recogntzethirds and halves and to under-stand their relationship to awhole.

B) Classroom work testsPractical testsTeacher observation

Me 52

Concepts

XVI. Subtraction-Investigating an algorism for

three-digit subtraction-Using an algorism for three-

digit subtraction-Using subtraction in application

situations


XVII. Liquid Measure-Using cups, pints, and quarts

to measure liquids


A) Teaching MethodB) Learning Activities

A) Show by use of the chalkboardsthat subtraction is the inverseof additionDiscuss problems orally in classWork problems step by step on theboardReview briefly the procedure touse when borrowing from tens place.Lead the children to seethat three-digit subtraction is basicallythe same as two-digit but nowthey must borrow from the hundredsplace.Explain the decimal point anddollar signEncourage the students to keep theirnumerals in straight columns orthe designated places,

8) Use play or real money in moneyproblems

Give much practice in writingsubtraction algorisms involvingmoneyGroup to meet specific needs ofstudents within the class.

Flach cards may be used for drillof subtraction factsUse worksheets pp. 23146 forpractice

A) Discuss liquid measurementHave students tell of experienceswith cups, pints and quarts

Bring in a pail of water. Fill thedifferent size containers and seehow much each will, hold.Help children discover how manycups in a pint- pints in quart, etc.

Discuss the larger quantity the cupthan pint, etc.

B) Permit children to become activelyinvolved in the measurilg activity

63

M(353

A) Printed

B) Audio VisualResources C) People

D) Places

A) Greater Cleveland Math ProgramTeacher's Manual pp. 218 -229G.C.M.P. worksheets pp. 231-246Key T(IDAS2j1Latilglic§,SPrPrimaxy Teachers,SRA

B) Money - real or play moneySale ads for problemsBundles of sticks20 bundles of 10020 bundles of 1020 single sticksNumber stripsCountingmenrlannel board, numeral symbols,and money symbols

A) Greater Cleveland Mah Prograft-1::Teacher's Manual pp. 230-232Dittos made by teacher

B) Containers:cupspintsquarts

Pail of water


A) Pupils know how to place numeralsin proper column for subtractionof three digits including dollarsand cents.Students demonstrate the abilityto compute three digit subtractionincluding the procedure used inborrowing from hurOrcd5 place.

B) Weekly testsSelf evaluationWorksheets po. 231-246Observation

A) Pupils become aware of theadvantage of having standardunits of liquid measure and willgain an understanding of quart,pint, and cup as standard unitsof measure.

Children see the relationship be-4tween the pint and'qilart, thepint and cup, and the cup andquart.

Students see the relationship ofliquid measure to daily living.

B) Oral testsObservation

64

?, e 54

Concepts

XVIII. Addition and Subtraction-Computing sums and differences-Extending underccanding of the

commutative and associativeproperties of addition

Time; approx. 5 days

A) Teaching MethodsB) learning Activities

A) Work sample problems at the board.Lead children to discover the

. commutative and associativeproperties of additionGuide students in P. class discussionof the commutative property - "thelaw of order", 34-2=5; 2+3=5Do likewise for the associativeproperty of-"law of grouping"(3+2)+4=9 or 3+(2+4)=9

B) Provide some meaningful activitiesfor children to gain better under-standing and practice in computingsums and difference. This maybe done through board work, games,ditto sheets or G.C.M.P. worksheetspp. 247-262

11 e 55

.6.) Printed

B) Audio VisualResources C) People

D) Places

A) Greatsr Cleveland Math ProgramG.C.M.P., worksheets pp. 247-262

B) CountingmenNumber line


A) Pupils indicate by their work andintelligent questions that theyare extending their understandingof the associative & ccamutativeproperties of addition

Pupils are able to apply theirknowledge of addition andsubtraction to compute sums anddifferences ones through

hundreds

B) Oral testsPupil observationAchievement testsDaily work

1,4 e 56

Concepts

XIX. Repeated Addition-Investigating repeated addition-Counting by 2's, 3's4's, & 5's.

Time: appro% 4-5 days

67

A) Teach4ng MethodB) Learning Activities

A) Use the flannel board and cut-outsto illustrate how equivalent setsmay be joined to finJ the numberof object' altogether. Lead thechildren to discover the repeatedaddition pattern in relation tocounting by 2's, 3's, 4's and S's

Have class discussion involvingthe many things in the child'senvironment which comes in 2'spair of mittens, socks, twins, etc.

B) Use groups of children or concreteobjects to divide into equal groupsof 2's, 3's, 4's, and 5's.Ask questions such as:

How many groups of boys?How many groups of girls?How many altogether?

Provide meaningful experiences forchildren to visualize the repeatedaddition in terms of setsExample: as children line up forrecess, gym class or lunch, have .

them walk in 2's, 3's, 4's, and 5'sCall on two sets to move out of theroom or line.Ask how many children;arc in this set (or group) Have

the next three sets step forth-how many in this set. Continuein like-manner to determinetotal number of children in setsof 4,5 and 0 sets.

" e 57



A) Greater Cleveland Math ProgramG.C.M.P., worksheets pp, 263-272Key Topics in Mathematics forPrimary Teachers, SRA

B) Flannel boardFelt cutoutsBooks, pencils, or other objectsPlastic numerals and symbolsBead frame

A) Expected OutcomeEvaluation 1') Testing Program

A) Pupils are able to count by2's, 3's, 4's and 5's

Students develop the readinessfor multiplication by investi-gating repeated addition

B) Diagnostic testsOral perticipatio:1Teacher observationDaily rEormanee on workshevtgSelf evaluation

68

Me 58

Concepts

Mathematics - Third YearReview approx. 3 weeks

I. The Product- Arrays

- Partitioning


II. Properties of Multiplication-Meaning of multiplication- A3 a sum-Equivalent products


III. lime-Hours- Minutes-Reading a clock- Seconds


69


A) Oral aiscussion and demonstration

of previous concepts taught.B) Provide practice sheets for review.

A) Explain meaning of array, product,and partitioning.Draw arrays on the board. Demon-strate how an array can be used tofind out "how many".

!3) Children draw arrays - Use thearrays to find out "how many".

A) Oral discussion and demonstrationof the commutative, associativeand distributive properties. (Itis suggested that these be dealtwith at a slow rate and atdifferent class periods.)

Guide stude.te through an investi-gation of the properties ofmultiplication.

B) Use concrete objects to seperateinto equal groups.

Make pictures of arrays todemonstrate points.

Develop chart with children toshow key words and symbols whichgive, clues to the multiplicationproperties.

Refer to Teacher'b Manual foradditional activities.

A) Review concepts of 1/3, 1/2, 114,through oral discussion andexamples on board.

Show "le relationship of the aboveconeb, s to time.

B) Let pupils take turns in settinghands of a clock to specifiedtimes given by various pupils.Students make up problems dealingwith time.Pupils make individual clocks withmovable hands to set at varioustimes.

M 59



A) Greater Cleveland MathematicsProgram Unit 1 (Teacher'sManual) Latest Editio',

G.C.M.P. text A 1-11B) Chalkboard or other visual aids

A) Greater C:Levelend Math ProgramUnit 2

G.C.M.P. worksheets L 11-20My Topics in Math 12r PrimaryTeachers, SRA

B) Material for chartMagic markerConcrete objects suTil as:Blocks, clothespins, beads, desks,cLildren themselves, etc.

Poster paper (cut into squares)Filmstrip:Commutative Property of Multi-plication, FOM

A) Greater Cleveland Math ProgramUnit 3G.C.M.P. worksheets A 21-24K1v Topics iri.ifath,Sgr,

TeachersB) Individual clocks (either made or

purchasedOne large clockFlsnnel boardFlannel cutoutsGame - Tell Time Quizmo

7


B) Diagnostic testing of previouslytaught concepts in basicaddition and subragtiOri.

A) The significance of an arraybecomes apparent to the students.Students are able to use arrays tofind products; thus understandingthe meaning of multirOination.

B) Oral testingObservatjJnSkills performed on worksheets

A) Students display the abili4 touse their concepts of theproperties of multiplication to

find the product.Are able to perceive the relatixship of multiplication toaddition.

B) Pupil observationDiagnostic tentsSkills demonstrated on worksheets

A) Pupils develop the ability to telltine accurately.

Pupils demonstrate the ability totell fractional measures oftime such as: 15 minutes' 1/4

20 minutes = 1/330 minutes = 1/2

B) Teacher made test duplicated orpaper.Teacher dictated tests of varioustimes.

Oral tests.

Me 60

Concepts

N. Numeration-Meaning of standard numeral-Place value

onestenshundredsthousands


V. Computation o7 Products-Repeated addition concept-Combination through 5 x 9

computation of one digitmultiplication


71


A) Oral presentation, explanation,and discussion.

Use of concrete and semi-concreteobjects to demonstrate.

Discuss dfatinction between numberend numeral.

Child demonstration.Board demonstration

B) Have children use blocks to builda cube. Guide students in aninvestigation of blocks used tobuild the cube. Help:them discoverthe multiples of tens pattern.Use countingmen to explore placevalue concepts.

Board races - Write numerals orcompute as teacher dictates,

Refer to Teacher's Manual foradditional help and activities.

A) Oral discussion and.eramples on..chalk board and flannel board.

Compare relationship of multipli-cation to addition.

Oral discussion of the meaning ofmultiplication sentences -5 x 240 ( 5 groups with 2 ineach group). This concept needsto be taught very carefully withmuch oral practice on behalf ofthe pupils.

Stress the equality in groups.Example: 5 x 2 (Each group willhave two).

1) Hake pictures to illuotrate facts.Permit pupils to manipulate variousconcre.e objects to separate intfgigul groups.Have students help. each other withflash cards.

Give written practice.Have children play game by placingball on a number and give theanswer.

Multiplication game.. : q e'61,



A) Greater Cleveland Math ProgramUlla 4

G.C.M.P. worksheets A 25-38Em_112,iss in M.th for PrilaryTeachers, SRA

B) Place value chartsCountingmenBundles of sticksVarious concrete objects whichmay be in classroom.Refer to Teacher's Manral foradditional list which may be usedbut not availabL, to each andevery teacher.

Filmstrip:Names for Numbers, FOM


G.C.M.P. worksheets A 39-60Key_Tordee_inillth-for PrbwarYTeachers, SRADitto sheets for practice

B) Pictures - chartsMato gameConcrete objects such as:beads, toys, books, clothes pinsFlannel board and symbol cutoutsUse chalkboardFlash cards

72

A) Expected OutcomeEvaluaticl, B) Testing Program

A) Pupils exhibit an understanding ofthe ones, tens, hundreds, andthousands place in our numerationsystem by their ability to usethe concepts in simple computation.

B) Oral tests'Progress made by students asobserved by teadher.

Teacher made test.Now well the btodents u'e theirskills on the daily work sheets.

A) Pupils demonstrate the ability toperceive the relationship betsseenmultiplicatimand addition.

Pupils are able to conip"te onedigit multiplication problemcthrough 5 x 10.

B) Diagnostic testsChalkboard testsDictationObservationSkills perforL,ed on worksheets

Me 62

Concepts

VI. Addition and Subtraction Facts-Commutative property of addition-Place value

adding and subtracting twoplace numerals.adding and subtracting threeplaca numerals.adding and subtracting fourand five place numerals.

-Using zero as a place holder inaddition and subtraction


VTI. Addition and Subtraction ofWhole Nurbers- Addition

2-3 aigit addition withcarrying to tens place.computation of 2-4 digit ad-dition with carrying to thethousands place.column addition of 3 and 4numerals and 4 digits.

-Subtraction3-4 digit subtraction withbormaing in ones, tens, andhundreds place.

computation of 2-4 digit sub.?.traction with borrowing.

-Addition and subtraction asinverse operations

-Linear measurementaddition of like units (suchas inches added to inches).Subtraction of like units

-Measurement of moneymonetary units to hundreddollars; using the period anddollar sign to express money.

-Reconstruction problems-Story problems

Timer approx. 11-12 days

.73


A) Oral presentation, explanation anddiscussion of place value,Use pocket chart showing ones, tens,hundreds and thousands. Usage ofsticks, bundles of paper strips,or countingmen will prove helpfulin stressing place value.

To show zero as a place holder usea chart.

B) Have children take turns going tothe board, chart or countingmenand illustrate place value as theteacher cells off numberb to beillustrated. When one child showsthe illustration, have anotherchild write the numeral thatcorresponds on ths board. Givemuch practice.

A) Use chalkboard, flannel_ board,charts, countingmen and otherrisual means to illustrate place -.value, regrouping, and the basicalgorisms for addition and sub-traction. It is suggested thateach of the concepts be taught atdifferent times. Much practiceshould be gi.zen dealing wiji eachnew concept; thus making way forfuture, accurate computation.

B) Seat games.. Board excereises and gameo for

class drill.Use concrete objects to showinverso relationship of additionand subtraction.

Stress legibility and straightcolumns.

Work reconstruction problcatogether as a group.Use problems arising in classroom,lunchroom, playground or any lifesituation to aid in linearmeasurement of money problems.

M e 63

A) PrintedB) kadio Visual



G.C.M.P. worksheets B 61-80Key Topics in Math for PrimaryTeachers, SRA

B) Place value chartsPocket chartsCounting sticksPaper stripsCountingmenRefer to feacherls Manual Unit 6for additional materials andteaching suggestions

Filmstrip:Two Properties of Addition


G.C.M.P. worksheets pp. 81-104Kev Tonics to Maya for19.2,92"le, SRA

B) Place value chartsFlannel board and cutoutsAbacusCountingmenToy and real moneyPocket chartsRulerYardstickFilmstrip:Place Value and Subtraction

74

A) Expected OutcomeEvaluation 2) Testing Program

A) Pupils demonstrate the ability towork with place values.Pupils are able to compute additionand subtraction of 2-5 numeralsusing zero as a place holder.

B) Use of knowledge in classroomsituations.

Ability to apply skills on workpages 61-80.

.Direct observation of the childrenas they are engaged in a learningactivity is the best appraisalmethod.

A) Pupils demonstrate the ability toapply tiieir knowledge of place

value, regrouping and inverserelationship, to compute additionand subtraction combinations of2-6 digits with both borrowingand carrying.Students express their understand-ing of the importance of keepingcolumns straight.

Children are able to apply theirknowledge of measurement to agiven situation.

B) Diagnostic testTeacher made test on each concepttvightObservationWorksheets on pp. 81-104

M e 64

Concepts

VIII. Computation of Basic Products-Verifying products by using

repeated subtraction-Product as a sum of partial

products-Computation of multiplication of

whole numbers


IX. Geometry-Comparative sizes cf rectangles

in relation to the number ofsquare units needed to coverthe rectangles.

-Area of a rectangle (informalintroduction).

Tine: approx. 2-3 days


A) Use flannel board and felt cutoutsto illustrate the partitioning ofsets. Review the term "partition".Present, discuss and illustratethe thinking process used infinding the product as a sum ofpartial products.Use chalkboard or flannel board toillustrate the concept of repeatedsubtraction as related tomultiplication.

B) Use flash cards for drill on thebasic multiplication facts.Play "multo" game.Board work by class members.

A) Review term rectangle. Use chalk-board or preferably a flannelboard and felt cutouts to demon-striae comparative sizes ofrectangles and squares. If possibleuse a large concrete block, box,or etc. and small square ones toshow the number of squat es neededto cover the large rectaligualrbox, block, etc. GlUde pwils indiscovering the area of a rectanglu.

B) Use geometric figures such assquares and rectangles in acreative way. Make a pictureusing different geometric figuresin art class.

Paper folding of squares,rectangles, and parallel lines.

M e 65



A) Greater Cleveland Math ProgramUrit 8

G.C.M.P. worksheets B 105-124Key Topics in Math for Prima*-vTeachers, SRA

B) Flannel board and array cutoutsMulto gameFlash cardsConcrete objects suoh as:Buttons, sticks, blocks, toys,etc, help wit.' factsYarn

A) Greater Cleveland Math ProgramUnit99

G.C.M.P. worksheets B 125-131Key Topics in PlItchforleachers, SRA

Flannel boardFelt cutouts of squares andrectangles

Ruled paper in squaresConstruction paper figuresFilmstrip:Experimenting With Area, FOR

76


A) Pupils are able to compute un-familiar products as the sum of

partial product.s.Students demonstrate the abilityto compute products through .

10 x 9.

B) Teacher made testDictationObservationDiagnostic tests

A) Stude:ts are able to determinethe number of square unitsneeded to cover a rectangulararea by using comparative sizesof rectangles and squares.

0 Teacher observationPup'l evaluation of his workWorksheets B 125-131

14 e 66

Concepts

X. Computation of Basic Products-Investigating properties of

multiplication (informalintroduction)CommutativeDistributive

-Partitioning arrays to computemultiples of 8 and 9

-Using arrays to compute products- Verifying products by using

repeated subtraction


XI. Numeration: Thousands-Writing standard numerals for

sums of thousands, hundreds,tens and ones

- ising commas to separate anumeral into periods

-Reading and interpretingnumerals for large numbers(mill:ons, billions, ortrillions); using expandednotation

-Using place value in computa-tion of sums and differences

- Using numeration and basicfacts to compute products oftens, hundreds, or thousands

-Interpreting standard numeralsin many ways

Time: approx. 2 weeks

77


A) Use arrays as models to show stu-dents how to compute products. Usea step-by-step procedure and askquestions as you illustrate thoprocedure.

B) Provide meaningful practice for thestudents through board work orappropriate ditto sheets.

Divide the class into groups. Giveadditional practice for those whounderstand readily or have thosestudents help pupils who need extrawork. Move about classroom givingneeded help to students as theywork their assigned work pages.Drill workMato game

A) Review the meaning of each word orintroduce new words relating toour numeration system. Example:million, thousands, etc. Do thisbefore you begin your presentationof each lesson.Review place value by writing thenames of the places on a chart orthe chalkboard.

Place a 1 in ones place and havethe numeral identified and itsmeaning given. Put a 1 in the tensplace and a 0 in the ones place.Repeat procedure as for ones. :Extend this activity to the. otherfour places.

Challenge the students to name anumber beyond 999,999 - discussmeaning and read aloud. Discussdifferent ways of thinking oflarge numerals such as millions.

B) Do much oral work with students tohelp them read large numerals.Send small groups to the board towrite large numerals.

He 670




G.C.M.P. Worksheets B 133-152Kev Tordeg in Math for Primary

1124qh2r2, SRAB) Flannel board

Felt array cutoutsFlash cardsMulto game


G.C,M.P. worksheets C 153172Egy Tonics in Matb12L2rkaaIochers, SRA

B) CountingmenNumeral cardsPlace value chartPaper, pencilFilmstrip:Names for NtAmorals

78


A) Students demonstrate the abilityto compute multiples of 8 and 9.

B) Self - evaluationTeacher made testObservationWorksheets B 133-152

A) Students show an understanding ofour numeration system.Pupils are ahle to write stande.rdnumerals for sums of thousandsand to use commas to separate anumeral into periods.

Students demonstrate the abilityto compute products of tens,hundreds, or thousands.

B) Pupils grade their own work,correct mistakes and bring workon pp. 153-172 for evaluation.

Weekly teacher made test.

M e 68

Concepts

XII. Long Multiplication-Finding products as the sum of

partial products-Distributive property-One place multiplier-Two place multipliers-Three place multipliers


A) Teaching MethodsB) Learning Activit3(.3

A) Discuss terms: multiplier, product,partial product, x, carrying,times, distributive property, etc.Use flannel ward or chalkboard topresent examples of work. Guidethe children carciAilly throughthe thinking process of the multiplication algorism

Make much use of the oclianded nota-tion to partition tie arrays.

The distributive ;Tol'erty should bereviewed b(:fore beginilire; as this

property s:aoul,] be u3eA in

explaining lellg multiplication.For ease of conputatiao, partialproducts may be arranced vertically,

Discus the need for keeping par-tial products in stral3ht verticalrows.

B) Pupils demonstrate the time savingvalue of multiplication.Pupils work individually accordingto their needs.Use board work for finding pupils'errors and helping them to makecorrections.

Progress from one to three placemultipliers as pupils are ready tomove.

Nave pupils work on worksheets pp.C 173-214. Have them work at theirown rate, use the teachers' manualto check their work, make cor-rections, and bring to teacher forevaluation of work. Give individualhelp as needed.

Divide class into 2-4 teams. Putproblems such AS 678 X 279 on theboard. Neve each member of theteam work a partial product -9 x 678, 7m. 678, 2 x 678. Thecaptains add sums to find products.

79

e 69

A) PrintedB) Audio 1/.7.str.1


A) Greater Cle-eland M-th ProgramUnit 12

G.C.M.P. worksheets C 173-214Key Tipics in Math for Primary

Teachers, SRAB) Chrilkboard

PaperPencilsFlannel boardArray cutouts

8


A) Students are able to apply theirskills in computing Jong multi-plication by making use of thedistributive property, expandednotation and partitioning arrays.

B) Observation of work on hoardTeacher made testsPerformance on worksheets C 173-214Diagnostic tests to determineareas of difficulty

Me 70


XIII. Areas of Rectangles-Finding ths area of a rectangle

as length times width-Problem solving

What happens to a rectanglewhen one side is doubled.

What happens to a rectanglewhen two sides are doubled.

-Investigating shapes to determinewhich to use for covering


XIV. Fractional Numbers-Linear measurement

Using fractional numbers tocompare lengths.Picture problems involvingfractional numbers.

-MoneyUsing fractional numbers tocompare value of coins.

Story problems involvingfractional numbers.


81

A) Review terms of rectangles andsquare units.

Draw examples on board. Explain howto find an area of a rectangle bymultiplying length times width.This should be done after muchpractice with covering rectangleswith squares.Use board work for practice, makescale drawings, and use gravhexamples.

B) Use floor or ceiling tile for visualaids.Have students do paper folding toaid in the concepts of squares,rectangles, parallel lines andparallelograms.

A) Discuss meaning of fractional numbersExplain what the top number meansand what the bottom number means.Example: 1. The "one" means we'retalking about one part of somethingthat has been divided or separatedi.to four parts. The "fours meanssomething that has been dividedinto four equal parts (or use fourgroups).

B) Have students practice writingfractional numbers on the board.Students make fractional parts ofa whole with construction paper.Use concrete objects in a set toshow the idea of a fractional partof a group.Provide opportunities for childrento measure things and write thecorresponding frictional number.Measur3 such things as books, desks,pencils, etc.Practice drawing different geometricfigures as teacher dictateb lengthto be drawn.Orally discuss and work out meaningful story problems involving 1

fractional parts of money.

M e 71



A) Greate... Cleveland Math ProgramUnit 13

G.C.M.P. worksheets C 215-2284ev Topics inyat_thE.2iLimaTeachers, SRA

B) Graph paperRulersPaper to foldFlannel boardFelt cutouts of large squares,small squares, large and smallrectangles

Floor and ceiling tileFilmstrips:Experimenting With AreaLines and Parts of Lines


G.C.M.P. worksheets C 229-244

1522/39121221171.11a2LUZUIGICI,Zgadtepa, SRA

B) RulersYardsticksTae measureMoney kit (pennies, nickels, dimes,quarters, half dollars)Number lineConstruction paperFraction kitPen:A.16

Books

82

A) Rxpected OutcomeEvaluation B) Testing Program

A) Pupils acquire a working knowledgeof and skill in finding areas ofrectangles.

Students realize that areas can bethe same in squares and rectangleswith different measurements.

B) Teacher observation of pupilprogressAchievement on work pages C 215-228Teacher judgment

A) Students are able to read andwrite simple fractions.Pupils develop the abili4 toapply their knowledge offractional Aumbers to everydaysituations.

B) Oral testsTeacher made quizzesDaily work

M e 72


XV. Missing Factor Multiplication- Problem solving

Missing factor problems.Completing multiplicationpuzzles.Story problems (involves in-formal division).

-Using arrays to compute products.-Commutative property

Rearranging factors does notchange the product.

Time: approx. 1 -2 weeks

XVI. Multiplication Shortcuts-Combining partial products to

shorten the multiplication .algorism.

-Computation drill of multipli-cation combinations using twodigits with two multipliers

- Computing products without usingpartial products

- Computing sums


A)

B)

Demonstrate how to use arrays asmodels to solve missing factorproblems.

Class discussion.Have students take turns workingexamples at the board.

Show relationship between partition-ing or dividing and finding themussing .factor. For example;? x 6 = 30 or 30 * 6 = 5. Usevisual means to point this out.

Have "better" students become"teachers" to help students whoneed lots of individual help.

Have students make up storyproblems and other students workthe problems.

A) Put examples of problems on theboard. Demonstrate how to combinepartial products to make the al-gorism shorter. Ask questions suchas listed on p. D 271 and 279 tohelp with understanding of process.Develop chart with children to showthe process used.Example: 36

x12, multiplyla (ones x ones)miq (tens x ones 4

ones x tens)1Z901 (tens x tens)1512

B) Provide many opportunities for stu-dents to do practice work, prefer-ably board work until the processis better understood.

Use flesh cards for games tostrengthen understanding of basicfacts.

Games such as Multo for learningmultiplication facts.

/

1 )

e 73



A) Greater .leveland Math ProgramUnit 15

G.C.M.P. worksheets D 245.270Key Topics in Math for PrimaryTeachers, SRA

B) Flannel boardFelt cutoutsNumeral °aidsArray cutouts


G.C.M.P. worksheets D 271-294

L9S19121k2111-ifiltbfarEakIlMTeacherq, SRA

B) Array CutoutsFlannel boardChalkboardMaterial to make chartMagic markerFlash cards for games

84


A) Pupils are able to apply arithmeticknowledge to real problem solvingsituations.

B) Teacher made quizTeacher observationHow well students apply theirskills to the everyday practiceworksheets

A) Pupils exhibit the ability totompute products of ones, tens,hundreds and thousands bycombining partial products andwithout using partial products.

B) Teacher judgmentTeacher observationWorksheets D 271-294Weekly quiz

M e 74

Concepts

XVII. Measurement-Liquid measurement

An investigation of thequantities used to describeliquid volume.Greater than - leas thancomparisons.

computing amounts of liquid.-Problem solving

Story problems involvingliquid quantities.

-Addition, subtraction, andmultiplication combinationsinvolving liquid measure

Time; approx. 1 week

XVIII. Application3-Problem solving

Story problems involvingproducts.

Story exercisis: sums anddifferences.

Computing costs.Computing amounts of liquid.



A) Demonstrate the use of the measures:pint, quarts, and gallons in class.Stress their comparative quantities,Put examples of problems on theboard. Work examples and discusstheir usage in practical ways.

B) Develop a chart with students,stating the comparative quantities.Example: 2 cups = 1 pint

2 pints = 1 quartVisit dairy farm, factory, fillingstation, etc. to see measures inuse.

Have students make up problemsdealing with liquid measures. Thenwork each other's problems.

A) Read each problem with the class.Guide the students in the thinkingprocess. Ask questions such as:1. What does the story tell?2. What does the problem ask?3. What operation should I use?4. What numerals are involved?5. How do I solve the problem?Work examples. Then ask:1. Is the answer reasonable?2. Check work.

B) Make chart listing the questionsabove. Encourage studentu to thinkcarefully about each phase of theof the problem.

Set up play store with empty cartetons, boxes, cans, etc. Price theitems with tagboard price tags.

Use clock to review time.

5

M e '?5)

A) PrintedB) .Audio Visual



worksheets D 295-306Egy_Topics in Math for Primaryleachers, SRA

B) Pint, quart, and gallon containersto be used for measuring liq'iidsMaterial to make chart


G.C.M.P. worksheets C 307-316Key topics in Math for PrimaryTeachers, SRA

B) Materials for chartMagic markerChalkboardClockMaterials for "play store"Price tags made from tagboard

Filmstrip:Solving Problems Which Operation?

86

A) Enacted OutcomeEvaluation B) Testing Program

A) Pupils acquire a working knowledgeof and skill in the use ofliquid measure.

B) Test accuracy in measuring ofpints, quarts and gallons.

Ability to apply their knowlec,seof liquid measure in practicalsituations.

Written tests.Teacher observation.

A) Students are able to applyprevious taught concepts tohelp them solve story problems.

B) Diagnostic tests to find area ofdifficulty (Plan to giveadditional work in area needed).

Worksheets D 307-316.Teacher obervation.Self evaluation.

M e 76

A). Teaching Methods.Concepts B) learning:Activities

XIX. Computation-Increasing computational A) Brief review of the basic operations

skills of addition, subtraction, andmultiplication.Assign a limited number of plgesto be worked after each lessonfor those students Ilho need day-to-day instructions. Then reteachand assign other pages. Giveindividual help as needed.

For students who are capable ofworking well on their own, assignthe whole unit. Have studentscheck their work by Teacher'sManual, correct their mistakesand come to teacher for additionalhelp if student missed more than2 or 3 problems per page or asneed arises.

B) Assign "teacher helpers" forslower students.

Encourage children to work on theirown, proof-read their work andtry to improve their computationalskills.

Me 17

87




Kri Topics in Math for PrimaryTeachers, SRAG.G.M.P. worksheets D 317-337

88


A) Pupils demonstrate the abilityto compute problems of addition,subtraction, and multiplicationwith a certain degree of accuracyand understanding includingmeasurement of money and linearmeasurement.Students are able to.think througha problem rather quickly withoutusing detailed step-by-stepprocedure.

B) Teacher made testWorksheets D 317-337Diagnostic test for the year'swork

Me 78


4 t

Mathematics - Fourth YearI. Sets and Problem Solving

II. Numeration

III. Rounding Numbers

IV. Arrays and Problem Solving

A) Help the students to r.!.ad athought problem and be able totranslate the problem into amathematical sentence.

B) Have the students discuss problemsand try to write them out onpaper.

A) This chapter deals with the inter-pretation of place value.

B) Have students read and discussmaterial. Solve exercises withteacher assistance.

A) Rounding of numbers is a basictechnique in making approximations,When estimating the answer for anexercise, it would be helpful toknow the minimum and maximum ofthe range in which the answerwill be found.

T;) Have students round numbers beforethey compute the results of aproblem.

A)Help students read and be able towrite mathematical sentences fromwritten sentences.

B) Students should solve mathematicalsentences on paper having masteredpreviously basic multiplicationand division combinations.

Me 79



A) G.C.M.P. Intermediate Unitspp. 1-10vol. 3



A) G.C.M.P. Intermediate Units

pp. 39-048vol. 1


A) All pupils should exhibit anability to write addition andsubtraction equations to expressthe relationship among threenumbers.

B) Administer suggested quiz at theend of this unit or substitute ateacher-made testing dovice.

A) Students may interpret a numeralin many ways.

B) Administer suggested quiz at theend of this unit or substituteteacher-made testing device.

A) Students should be able to exhibitan ability to round a wholenumber to the nearest ten bychoosing the multiple of ten thatis closest in value to the numberwe are rounding.

B) Administer suggested quiz at theend of tha unit or substitute ateacher-made testing device.

A) Students should be able to demon-strate the meaning of multiplica-tion and division by being ableto state the number structure ofa atoey in equation form and thento use.

B) Administer suggested quiz at thoend of this unit or substituteteacher-made testing device.

M e 80

Concepts

V. Multiplication of Whole IlLabers

VI. Open Sentences

VII. Exploring Geometry

VIII. Measurement

91


A) Develop a readiness for the multi-plication algorism by usingarrays.

B) Have students solve multiplicationexercises on board and on paperwith teacher assistance.

A) Explain to the students the differ-ence between open and closedsentences. Help them to createtrue sentences from open sentences.

B) Unit can be done independentlyafter discussion on materiU hasbeen presented.

A) Discuss with the students the con-cept of a point. After some explo,..ration of the idea of a point and:locations of points, we begin aninvestigation of two points. Inconsidering two pointo, we seethat there is a distance betweenthem called a. line segment.

B) Have students read and discussthe unit.

A) Suggest that objects beingmeasured are compared with astandard unit.

B) Have the students investigate thedifferent methods of measuring.Use a balance scale in showingthe measure of weight.

H e 81


Resources 0) PeopleD) Places


PP. 49-70vol. 1


A) G.C.M.P. Intermediate Unitspp. 91 -105vol. .1


B) Balance scalesContainers (cups, pints, gallons)

92


A) Illustrate an ability to find theproduct of two whole numbers byemploying the basic multiplicationfacts.

B) Aaminister suggested quiz at thaend of this unit or substituteteacher-made testing device.

A) Students should display a skill inrecognizing a mathematical sen-tence to be true, false, or open.Also students should be able toconstruct true sentences fromopen sentences.

B) Administer suggested quiz at theend of this unit or substituteteacher-made device.

A) Students should dispy a clearnotion of a point al should haveperceived some of thl. Manyrelations between points.

E) Administer suggested quiz at theend of this unit or substituteteacher-made testing device.

A) Students should explore liciaid-lasurement and weight by empha-sizing the value of standards of

measuremsnt.B) Administer suggested quiz at the

end of this unit or substituteteacher-made testing device.

Me 82

Concepts

IX, multiplication of Whole Numbers

X. ..,issing Factor Problems

XI. Division Algorism

XII. Geometry: Lines

33


A) Review the multiplication algorismand demonstrate multiplication ofthree and four place numbers.Make students aware of theimportance of place value.

B) Solve multiplication problems onchalkboard and on paper.

A) Any two whole numbers have aproduct. In this unit the inverseproblem is investigated. Given aproduct and a factor, find theother factor.

B) Have students solve problems inthis unit on paper with teacherassistance.

A) Develop the division algorism forthe students. Use the equationb = aq + r to begin the idea offinding a quotient (q) and aremainder (r).

B) Have the students perform exerciseson the Chalkboard.

A) Explore the properties of lines.Help the studcats discover themeaning of skew, parallel, andintersecting lines.

B) Have students use str!ng todemonstrate kinds of lines.

Me 83

CI

A) PrintedB) Audio Vitalal

Resources C) People .

D) Places



A) G.C.M.P. Intermediate Unitspp. 157168vol. 3


A) This unit is designed to strengthenthe students ability to computeproducts.

B) Administer suggested quiz at theend of this unit or substituteteachermade testing device.

A) This unit provides students withthe opportunity to construct andsolve multiplication equationswith missing factors.


A) Students should exhibit the use ofthe dividion algorism.



B) String

A) Students demonstrateof lines.

B) Administer suggestedend of this unit orteacher-made testa

a knowledge

quiz at thesubstitute

, device.

94

H e 84


XIII. Money

XIV. Introduction to FractionalNumbers

XV. More Division of Whole Numbers

XVI. Dort.nal System

95

A) Review the value of differentUnited States coins. Show howeach amount lees than a dollarcan be written two way.. . Solveproblems using the fourfundamentals P-d money.

B) Have students ..ve problems indi-vidually on paper and on chalk:,board.

A) !felp students get a broader under-standing of fractions by use ofa number line and regions.

B) Solve problems in addition andsubtraction with fractionalnumbers.

A) In this unit, continued explorationof the division algorism is done.Pupils investigate patterns indivision equations to find whathappens to the quotient when onevariable remains the same and theother is increased or decreased.Emphasis in this snit is on accu-rate computation in using thedivision algorism.

B) Have students solve problemsindividually on chalkboard andpaper.

A) The decimal system of numerationis a place value system based ontens. The dozenal system ofnumeration is a 'lace value sys-tem based on twelves. Added co7,-putational skill in division by12 is accomplished in this unit.

B) Have students solve problems in )

this unit on board and paperwith teacher assistance.

M e 65



A) G.C.M.P. Intermediate Unitspp. 176-185

vol. 3B) U.S. coins

A) G.C.M.P. Intermediate Unitspp. 18t-211vol. 3


A) G.C.A.P. Intermediate Units

PP. 228-234vol. 3

DC


A) Students relate the names ofU. S. coins and bills to thequantities of money theyrepresent.


A) Students demonstrate a basicability to add and subtractfractional numbers having com-onuerominators.

13) Administer suggested quit, at theend of this unit or substituteteacher-made testing device.

A) Students display a more refineduse of the division algorism.

B) Give suggested quiz at the endof this unit or substituteteacher-made testing device.

A) Students illustrate a knowledgeof computation in the context ofthe dozenal system.


M e 86

A) Teaching Methods

Concepts B) Learning Activities

XVII. Linear Measurement

XVIII. Bar Graph and Histograms

XIX. Whole Number Computation

3

97

A) Standards of measurement of lengtlare discussed in this unit.Pro-blems of conversion from one unitto another using both fractionalnumbers and whole numbers arepresented. This unit provides agood review in addition offractions and mixed numbers.

B) Have the students solve exerciseson the chalkboard and on paper.

A) The dLfficulty in understuidinga large number of measurementsis primarily on of organizingthem so that the reader canquickly see the whole set ofmeasurements. To help describethe distribution, two statisticalconcepts are introduced - themean and the range.

B) Have students draw histograms onpaper with teacher assistance.

A) Increased skill in the computationof whole numbers is the object ofthis unit.

B) Help students improve in thisarea bo,r. practice on the board

with teacher guidance.

Mc S7




pp. 235-264vol. 4



pp. 274-298vol. 4

ti 8


A) Students establish a standard :forcomparison called a unit ofmeasure.


A) Students display an ability toshow a distribution by bar graphsand histograms.

B) Give suggested quiz at th,3 endof this unit or bubstituteteacher-made testing device.

A) Students exhibit a skill incomputation in whole rrmlbers ofsums, differences, products, andquotients.


He 88

A) Teaching Methods


XX. Exploring Geometry

XXT. Primes And Composites

99

A) This chapter is the third of atrilogy in which students first,investigate points, then lines,and now planes.Emphasis should be on the pupils'discovery of the relationshipsbetween two planes and amongthree or more plane. They shoulddevelop an ability to perceiveplanes suggested by physicalobjects in his environment andto perceive planr.s that do rothave physical models but areonly suggested by three psicalpoints.

B) Have students use cardboard tohelp develop a feeling theplans, but point out that card-board is only a model for asegm(nt of a plane.

A) Understanding of multiplicationwill be de-pened considerablyby an investigation of primeand composite numbers.

B) Students solve unit with teacherguidance.

M e 89




B) Cardboard


Pp. 307-312vol. 4


A) Students through investigatimfind a relation of lines andplanes in space.


A) Students disclose a knowledgeof prime and composite numbers.

B) Give suggested quiz at the endof this unit, or substituteteacher-made testing device.

U e 90

A) Teaching Methods


Mathemathics-Fifth YearI. Set of Quotients

iI. Open Sentences

rntv.,..tn<tino to FractioltAl NumbeLa

101

A) Number lines and regions are usedto show fractional numbers asquotients. Show these demon-strations on the chalk board.

B) Have students draw regions & numberlines to show fractional numbersas quotients

A) Demonstrate to the students how touse the number line Co determinethe results of an equati'-Example: 3X =12

3X and 12 both

appear at the same location onthe number line. From this rightend point decrease 3X to OXwhich will be the left end point.Divide the value 12 into thenumber of partitions the linesegment was divided into

12/3 24/30 . . 12

OX IX 2X 3X

By observation, X= 12/3 of 4

B) Have students work in small groupswith teacher to help in solvingproblems.

A) we consider the whole-numberquotient firat,then fractionalnumber quotients. This chapterdevelops set comparison as abasis for understanding themeaning and use of fractions.

B) Have students work independentlyor in small groups with teacherguidence.

He 91


Resources C) keopleD) Places

A) G.C.H.P. Intermediate Unitspp. 1-13vol. 5



102


A) Students present quotient asfractional members

B) Administer suggested auiz at theend of this unit or substituteteacher-made testing device

A) Students will continue thedevelopment with open sentencesthat define open sentences

B) Administer suggested quiz at theend of this unit or substituteteacher-made testing device

A) Students develop set comparison asn 4,1Hs f,,r understanding '.he

i g and use of fractions

P) AJ..inister suggested quiz t theend of this unit of substitutetescher-mad,e testing deviee

Me 91

A) Teaching Methods

Concepts B) Learning ALtivitie:,

IV. The Properties of the Set ofFractional Numbers

V. Mcltiplication of FractionalNumbers

Vt. nivotiou of Frnotiounl Numbcts

103

A) Show in this chapter that theproperties of the set of wholenumbers apply to the set offractional numbers.

The set of fractional numbers hasone property that the set ofwhole numbers does not-theinverse property for multipli-cation. For each fractionalnumber a/b (except 0) thereexists a fractional number b/asuch that a/b x b/a=1

B) have thestudents use this inverseproperty or reciprocal idea tosolve problems in this unit

A) Show the principle of multiplicationof fractions by shaded regions.Draw these regions on the chalkboard to illustrate this idea.Develop from this the processof finding the product bymultiplying the numerators andthe denominators together toobtain the final numerator and

denominator.

B) Have the students draw regions tofind products.

A) Show division of fractions bycreating a complex fraction (tomthe given division problemsClear the fraction by multiplyingthe numerator and denominatorboth by the reciprocal of thedenominator. Thus, the prcblembecomes a multiplication problem.

B) Have students try to work throughthe logic behind the solutionof division of fractions inproblems

Me 93






104


A) Students display the knowledgethat the properties of wholenumbers apply also to fractionalnumbers


A) Students display an ability tocompute products in the set offractional numbers


A) Students display an ability tocompute quotients in the set offractional numbers.


M e

A) Teaching Mett,odsConcepts B) Learning Activities

VII. The Bux

VIII. Fractional NumbersAddition & Subtraction

X. Geometric Figures

X. Measurement: Area

A) In this unit we explore the twenty-eight lines determined by theverticies. Any two lines canbe seen to be parallel, inter-secting or skew

B) Have students c)nstruct a rectangularprism for their use in solvingthe ideas of this unit

A) Sturlents have already acquired somefamiliarity with the additionand subtraction of ErLetionalnumbers whose fraction' havecommon denominators. In this

unit we will investigate theproblem of adding fractionalnumbers Whose fractions havedifferent denominators.

B) Have stude:Ics do computations onthe chalk board and at theirdesks.

A) Help the student to be able todetermine the amount of face,edges and vertices of the figuresin this unit.

B) Have student work in sm.111 group orindividually Witt etherguidance.

A) Show the students the importance ofa standard unit of measure fordetermining area. Developformulas Eor finding the area ofa rectangle. Use this idea incomputing t:, area of compositeareas.

B) Have students in small groups dote;mine the area of a compositefigure.

Me 95



A)G.C.M.P. Intermediate Unitspp. 93-104vol. 6



B) Model ofTriangular pyramidTruncated pyramidOctahedronTetrahedronCube



A) Students investigate points, lines,and planes in a rectangularprism


A) Students should acquire a skill incomputation of the sums anddifferences of fractional

numbers


A) Students should investigate twoand three dimensional gedmetricfigures and be alile to idenEkEyfaces, edges, and vertices ofthese figures.


A) Students nompute the area of sometwo dimensional figures


Me 96

A) Teaching Methods


XI. Trial and Success

XII. Exact Division

XIII. IntrOductiOnto the setOf Integers

XIV. Computation in the Set ofIntegers: Addition andSubtraction

107

A) Trial begins with a guess at thesolution of an exercise. Thetrial tells us whether theguess was too large or too small.This information then helpsus in making a better guess.Continued trials lead to a guessthat is the solution to the

exercise. This is success.

A) In the set of whole numbers, exactdivision is possible if thedivision can be multipliedby some whole number to obtain

the dividend.Practice is given in this unit to

long division of whole numbers.

B) Have students work individuallywith teacher help. Board workmay prove profitable for the

class.

A) In the set of whole nflmhers,addition is always possible butsubtraction sometimes caonot bezdmpleted. Thus, anew set of0mbers is created; negative and

positive whole numbers are called

integers.

B) Have students solve problems inthis unit with 'eacher help.

A) After the students have gottena basic ability to compute ad-dition and subtraction problemsusing the set of integers,continue solving addition andsubtraction in this unit.

B) Have students compute results toaddition and subtraction problemsusing the set of integers

M e 97





A) G.C.M.P. Intermediate Unitepp. 201-219vol. 7


108

A) Expected CutcomeEvaluation B) Testing Program

A) Students compute results to storyproblems by use of trial andsuccess method.

B) Administer suggested quiz at theand of this unit or substituteteacher -made testing device

A) Students compute exact quotientsto division problems


A) Students, by using the idea of thenumber line are able to exhibita knowledge of what integers are


A) Students compute the results toadeition and subtraction of theset of integers.

B) Administer suggested quiz at theend of this unit nr substituteteacher. made testing device

He 98


XV. Geometry: Angles

XVI. More about Integers

XVII.Measuring Angles

XVIII. Round Numbers

XIX. Scale Drawing

109

A) Discus:; the formation of an angleas the intersection of two raysSuggest ways of naming angles

B) Have students read and solve unitind?pendently with teacher help

A) Extend the knowledge of integersby applying physical models tothen in order to make additionand subtraction more ccncrete

B) Students solve unit independentlywit4 teacher help.

A) Demonstrate how to use a protractorto Eind the measure of an ar;le.

B) Have students use protractors tofind the measure of angles

A) The nunber line is a convenientdevice to explain approximations,

In checking the results of the exercise, it may be helpful to knowa greatest possible result anda least possible 7es7:1t.

B) Have the students round off numbersand approximate ri,sults, thencalculate answer. Check one

with another.

A) A scale drawing of an object is adrawing in which each lengthshown corresponds to a length ofthe object.

A scale drawing may be larger thanthe object, smaller than theobject, or the same size as theobject.

B) Student should practice drawingfigures to scale

M e 99



t) G.C.M.P. Intermediate Unitspp. 246-256vol. 7


A) G.C.M.P. Intermediate Unitepp. 271-287vol. 8

B) ProtractorStraightedge


A) G.C.M.P, Intermediate Unitspp. 303-315vol. 8

1.10


A) Students investigate the formationof an angle and the various kinds


A) Students increase their ability tocompute with integers


A) Students compute the measure of anangle


A) Students exidbit the ability toapproximate numbers may beused to determine approximateresults to mathematical exercises


A) Student illustratesa drawing toscale


Me 100


XX. Number structure of storyExercises

XXI. Introduction to Decimals

XXII.Computation

A) Show children a relationship ofword sentences & number sentencesHelp them to interpret words intonumber ideas

B) The children work individuallyand in small groups.

A) Expose the students to the ideathat fractions can be expressedas decimals. Help them discOverplace value as related todecimals

B) Have the students solveindividually Orwith teadherhelp problems in this unit

A) This unit is a review of thefundame tals dealing with wholenumbers, fractions, and integers

B) Have students solve this sectionindividually

M e 101


Resot:rces C) PeopleD) Places



A) (i.C.M.P. Intermediate Unitspp. 347-364vol. 8

112

A) Expected OutcomeEvaluation B) Testirg Program

A) Students exhibit the ability toexpress story exercises asequations or inequalities


A) Students demonstrate a knowledgeof the basic meaning of decimals


A) Students solve whole number andfractional number problems.


Me 102


CURRICULUM GUIDE

MATHEMATICS

MIDDLE

113

A) Teaching Methods


llathematics - Sixth YearI. Decimal Numeration

II. Addition and Subtraction WithDecimals

311. Multiplication With Decimals

14

A) Explain to the students by demon-stration and discussion the ideaof place value.

B) Have individual pupils working atthe chanboard. Use the multipli-cation form (for example: 400 =4 hundreds = 40 tens = 400 ones).Have students, in as many ways alpossible, interpret a decimal asa whole number of decimal parts.For example: 12.00 may be inter-preted as 1200 hundredths, 120tenths, and 12 ones.

A) Discuss with students the impor-tance of place value in theaddition and subtraction ofdecimals.

B) Have students work together andindividually in solving problems.

A) Discuss with students that tocompute the product of twonumbers, .7 and .6, first computethe product of the whole numbers,1 and 6, and then the product ofthe decimal parts (tenths xtenths).

B) Have students solve problems ofmultiplication on chalkboard andon paper. Have students workindividually and in small groups,

Mn r 1





A) Expected OutcomeEvaluation P) Testing Program

A) Students display a basic know-ledge of decimal numeration.

B) Administer suggested quiz at theend of this unit or substituteteaches -made testing device.

A) Students show ability to computeresults to addition and sub-traction problems with decimals.


G.C.M.P. Intermediate Unite A) Students compute results topp. 27-40 problems of multiplication withvol. 9 decimals.


M In 2

. 115

Concepts

IV. Division With Decimals

V. Decimals and Fractions

VI. Measurement of Volume

VII. Geometric Construction

118

A) Teaching MethodsB) Learning Activil,ies

A) Discuss in a similar way as withmultiplication of decimals. Forexample: .65 4 .5, first computethe quotient of the whole numbers65 + 5 and then hundredthstenths. The quotient of hundredthsand tenths may be explained manyways. Explanation by fractions isone way: 1/100 t 1/10 = 1/10.

B) Have students work at the chalk-board to check on their progress.

A) Show the relationship of decimalsand fractions by demonstration onchalkboard.

B) Practice in computing in bothfractions and decimals should bedone on board and individuallyon paper.

A) Explain and demonstrate to thestudents,using models,thereasoning behind finding volume.

B) Have students make models of thedifferent prisms studied.

A) Present this unit by use of alearning packet.

1) Students work independently andat their own rate.

M to 3





A) G.C.M.P. Intermediate Unitspp. 79 -305

vol. 10B) Cardboard models of:

rectangular prismstriangular prismshexagonal prisms

Wooden inch cube blocks

A) G.C.M.P. Intermediate Unitspp. 106-119vol. 10Learning packet on construction

117


A) Students compute results toproblems of division withdecimals.


A) Students are able to comparefractions and decimals as tosize.

B) Administer suggested quiz at theend of this unit or substituteteacher -made testing device.

A) Students investigate Volumesand surface areas of three -

dimensional figures.B) Administer suggested quiz at the


A) Students display a knowledge ofbasic geometric oonstruction.

B) Administer suggested quiz at theend of this unit or substituteteacher -made testing device.

M al 4

A).Teaching MethodsConcepts :13) Learning Activities

VIII. Metric Measurement

TX. Repeating Decimals

X. Histograms

XI. Integenot Addition and Sub-traction

118

A) Introduce students to the basicidea that the metric measure isset up on a basis of 10.

B) Have students measure objects withtheir metric rulers.

A) This unit gives added computation-al skills in division of decimals.Most quotients do not compute tobe even; therefore, the idea ofannexing zeros to the dividendis used to produce an evenquotient or to see that thequotient becomes repeating.

B) Have student work independentlyor in small groups computingquotients.

A) Demonstrate on chalkboard how toconstruct a histogram.

B) Have students construct on paperproblems in this chapter onhistograms.

A) Review with students the ideas ofadding and subtracting of integersas presented in Book 7, Unit 12.

B) Have students solve this unit atthe chalkboard and at their desks.

Mm 5




B) Metric rulersMeter stick



PP. 147-156vol. 10


PP. 157-166vol. 11G.C.M.P. Intermediate Unitspp. 190 -213

vol. 7

119


A) Students investigate the basicunit of the metric system.


A) Students are able to computequotients in decimal form bywhole number computation anddecimal part computation.


A) Students present information abouta group as a whole using a bargraph or histogram.


A) Students display an increasedknowledge of the use of integersin ai!lition and subtraction.

B) Administer suggested cuiz at theend of this unit or substituteteacher made testing device.

M m 6

A) Te-thing Methods

Concepts 13) Learning Activities

XII. Integers: Multiplication andDivision

XIII. Recipmcals of the Integers

XIV. Review of Fractional Numbers

XV. Fractional Numbers: Addition

,20

A) The students already understandmultiplicationAnd division ofwhole numbers. Therefore the signof the product or quotient willbe the new concept to be taughthere. This concept can be shownby the use of the number line.

B) Students can do most of theirwork individually with teacherguidance.

A) Explain to the students that thereciprocal of an integer "a" isthe missing facto: Px" in theequation ax = 1.Show students how to use the ideaof reciprocals in solvingequations for the unknown such

as 4x = 12.B) Have students solve problems in

small groups with teacher help.

A) Continue use of reciprocals insolving problems with unknownsequal to fractional numbP.s.

B) Have students solve problems onpaper individually and 1.n smallgroups with teacher help.

A) Show students Low to compute aleast common multiple ordenominator by the prime factormethod.Solve addition problems offractional numbers by finding aleast common denominator.

B) Students can solve this unitindividually on paper.

Mil 7






A) Students display an ability tocompute problems of integersin multiplication and division.

B) Arthilnister suggested quiz, at the


A) Students solve for the missingfactor of problems in the setof rational numbers.

B) Administer suggested quiz at theend of this unit or substituteteacher-made device.

A) G.C.M.P. Intermediate Units A) Studerts develop skills in

PP. 184-197 computation of fractional

vol. 11 numbers.B) Administer suggested quiz at the


A) G.C.M.P. Intermediate 'hitspp. l98-206vol. 11

121

A) Students exhibit an ability tocompute in addition andsubtraction of fractional

numbers.R) Administer suggested quiz at the


M m 8

') Teaching Methods


XVI. Rational Numbers as Differences

XVII. Rational Numbers: Computation

.VITT: Geometry: Spheres and Circles

122

A) This unit uses the fact thatrational numbers can be the re,-mainder of two fractional numbers,

B) Have students solve problems onpaper with teacher guidance.

A) This unit gives extra practice inthe computation of the fourfundamentals using integers,solution of equations, andcompletion of number sentences.

B) Have students compute problems onchalkboard and paper withteacher help.

A) Explore the ideas of radius,circumference, diameter, greatcircle, and area with thestudents. "se figures to showgreat cii ,e. Use drawings toillustrate other ideas.

B) Have the students solve theproblems in this unit on paperor on the chalk board.

Mm 9




A) G.C.M.P. Intermediate Units A) Students demonstrate that all

pp. 207-213 rational numbers can be

vol. 11 expressed cs aclifferealce oftwo fractional numbers.



pp. 214-234vol. 11


PP. 235-247vol. 12

B) SpheresCompassStraightedge

123

A) Students exhibit an ability tocompute in the set of rational

numbers.B) Administer suggested quiz at

the end of this unit or sub-stitute teacher-made testingdevice.

A) Students exhibit an understandingof spheres, circles, and theirparts and relations.


M ra 10

Concepts

XIX. Latitude and Longitude

XX. More About Primes

XXI. Introduction to Percentage -Part I

124

A) Teaching Methods.B) Learning Activities

A) Use this unit for enrichment astime allows.

B) Have.studentslwork independentlyon this material.

A) Re-examine the ideas of primenumbers. Solve a couple ofexamples on the chalkboard.

B) Have the students solve problemson the chalkboard and on paper.

A) Practice changing fractionsfrom hundredths to lowest termsand vice versa.Use ideas learned in Unit 13about reciprocals in computingresults to equations.

B) Have students solve problems onpaper and chalk board.

Hai 11




PP. 248-255vol. 12

B) Globe

A) G.C.M.P. Intermediate Unitspp. 256262vol. 12


125


A) Student display a knowledgeof the system of latitude and1c2igitude used to locate anypoint on the earth's surface.

B) Administer suggested viz at theend of this unit or substituteteacher-made testing device.

A) Students reveal the definitionsof primes and composites byreviewing primes and composites.


A) Students display the underlyingconcepts.of percentage withoutusing the language ofpercentage.


Mm 12


XXII. Introduction to Percentage -Part II

XXIII. Uses of Percent

XXIV. Properties of the RationalNumbers

126

A) Set up basic way of solvingequations that are in thefor of r% of B = P. One goodmethod of solving for r andB is by the usa of reciprocals.Solutions for P can be done bystraight multiplication.

B) Have students read a problem andbe able to determine whichnumbers can be substitutedfor r, B, and P.

A) In this unit the students setup equations in order to solvethem for mark up, cost, markup percent, selling price, andselling price percent. Checkover basic equations withstudents. Help them to seehow to substitute values.

B) Have students work in smallgroups with teacher help.

A) This unit deals with the com-parison of rational numbers asto size. Explain to studentsthat a common name forfractions and decimals shouldbe arrived at first in orderto make the comparison.

B) Have students visualize fractionsand decimals on a nuMber linein order to compare them.

M in 13




A) G.C.M.P. Intermediate Unitspp. 282-294

vol. 12


127

A) Expected OutcomeEvaluation B) Testing Progra'

A) Students exhibit the language ofpercentage in solvingfundamental problems.

B) Administer suggested quiz at theend of this unit or substitutetelaer -made testing device.

A) Students compute problems con-cerned with percent.


A) Students explore the orderproperties of the set ofrational numbers.


n 14

Concepts

Mathematics-Seventh and Eighth yearsI. Counting to Computation

1. Telling how many2. You and research3. The invention of number symbols4. The number and the numeral5. Egyptian numeral system

Supplemental materialEvaluation material

6. Babylonian numeral system7. Roman numeral system

Supplemental materialEvaluation material

8. The Hindu-Arabic numeral system9. Fractional and Decimal notation

10. Powers and exponentsEvaluation material

11. Other numeral systemsSupplemental material

12. Translating into base ten13 Translating from base ten

to other basesEvaluation material

14. Addition in other bases15. Subtraction in other bases

Evaluation materialSupplemental material

16. Multiplication in other bases17. Base two system of numerals18. Base twelve system of numerals

Evaluation materialSupplemental materialProjects for part I

128

A) Teaching Methods

8) Learning Activities

A) Units 1-11Conduct oral discussions

dealing with the developmentof our numeration system fromcaveman to present.Units 12-18

1cplain and demonstratenumeration systems that arebased on a system other thanten.

B) . Have students build forproject an abacus that can beconverted from one base to another.

Have students give demon-strations of the problem solvingtechniques involved in the basicoperation.

Have students individuallyand in groups set up problemsand solve them.

M m 15



A) Advancing in Mathematicspp. 3-46

B) Abacus

1 29


A) Students should develop anappreciation of how our numerationsystem was developed

Students should be able tocompute in bases other than base10 the operation of addition,subtraction , and multiplication.

8) Administer suggested quiz atthe end of this unit or substituteteacher-made testing device.

Min 16

Concepts

II. Exploring Sets and Operations1. Sets of numbers2. Universal sets

Evaluation material3. Well-specified sets4. Operations on sets

Evaluation material5. Closed operations6. Time out to talk about equality

Evaluation material7. The Commutative property8. A multiplication pattern9. Sets of factors

Evaluation material10. Multiplying three or more

numbers11. Difterent operations12. Even end odd numbers

Supplemental Material13. Prime factorization

Evaluation material14. Common factors15. Multiplication material

Evaluation materialSupplemental materialSupplementary problems forParts 1 and 2


A) Units 1-5By discussion and demonstration

show how each of the new symbolsare used with the involvementof sets.

Use practical examples todemonstrate union and intersectionwith sets.

B) Students should solve problemsin text and be able to sharereasons for decisions

III, Mathematical Systems A) Eiplain and demonstrate by '1. Round and Round-A new relation board work the ideas expressed2. A miniature mathematical system in modular arithmetic3. Circular addition

Evaluation material4. The identity element5. Circular multiplication

Evaluation material6. Inverse elements7. Circular subtraction and

divisionEvaluation material

8. The need for new elementsEvaluation materialSupplemental material

130

B) Have students on paper makea copy of the face of a clock.Use a paper arrow fastened to theface of the clock with a round-headed pin to represent thehour hand.

M rrt 17


Resources C) PeopleU) Places

A) Advancing in Mathematicspp. 49-90vol. 1 teachers' manual


Ditto illustration of a clock

A) Expected OutcomeEvaluation B) Testing 2rogram

A) Students should be able tc,display a good working knowledgeof basic sets.

Students should exhibit adeeper knowledge of multiplication.

B) Administer sugEested quiz atthe end of this unit or substituteteacher-made device.

A) Students develop br contrasta more meaningful understandingof properties of numbers.

Students broaden the conceptof a mathematical system and itsingredients-a set of elements, atleast one relation, and at leastone operation.

B) Administer suggestet. quiz atthe end of this unit or substituteteacher-made testing device.

31Mtn 18

Concepts

IV. Extension of our Numeral System1. The struggle to invent new .

numbers2. Figuring how much

Evaluation material3. Comparing fractional numbers4. Addition of fractional

numbersEvaluation material

5. Subtraction of fractionalnumbersEvaluation material

6. Division of fractionalnumbersEvaluation material

7. Multiplication of fractionalnumbers

8. Another look at division9. Fractional numbers in

decimal parts10. More about decimals

Evaluation material11. Percent12. Matching numbers with points

V. Places and Shapes1. Elements of geometry2. Geometric lines

1 Evaluation material3. Geometric surfaces4. Geodesics

Evaluation material5. Intersection and Union of

sets and pointsEvaluation material

6. Angles7. Triangles8. Curves

A) leaching MethodsB) Learning Activities

A) Discusa and explain theprocesses involved in solutionof the four fundamentalprocesses with fractionalnumbers.

Review material on decimals

B) Have students work ingroups and independently is thecomputation of fractional numbers.

A) Introduce students to thenotion that the elements ofgeometry are ideas and thatgeometric ideas can be repre-sented by physical interpre-tation.

B) Most of this unit will bedone individually by readingindependently.

132 M m 19





133


A) Students develop a deeperawareness of fractions.

Students extend their skillwith working with fractions byadded computation in the fourfundamentals.

Students develop a betterunderstanding of the relationshipbetween fractions and decimals.

B) Administer suggested quiz atthe end of this unit or substitiueteacher-made device.

A) Students should develop theconcept that space is the set of

all points.Students should be able to

exhibit a basic knowledge of lines,points and angles.

B) Administer suggested quiz atthe end of this unit or substituteteacher-made Testing device.

Mm 20

Concepts

VI. Mathematical Sentences1. Language called mathematics2. Constants and unknowns in

mathematical sentencesEvaluation material

3. Poodle problem in Pomerania4. Inequalities and graphing5. Another way of comparing

numbers6. Putting ratios to work7. Problem situations and

mathematical sentencesEvaluation material

8. Comparing numbers graphically

VII. Measurement1. What is measurement?2. an selects standard unit

measures3. Accuracy of measurement4. The Metric System of

measurementEvaluation material

5. AnglesSupplemental materialEvaluation material

6. Propettieg of triangles

and quadrilaterals.Supplemental material

7. Measurement of regions8. Circles

Evaluation uaterial9. What's inside

VIII. The Rational Nuders1. The negative numbers2. Addition of rational numbers3. Subtraction of rational

numbers4. Multiplication of rational

numbers3. Division of rational numbers6. Solution or truth sets

134

A) Teaching MethodsB) Learning Activitie3

A) Read and discuss material onmathematical sentences . It is

important that students under-stand that only when an unknownin a sentence is replaced by anumeral do we obtain a statement.The resulting statement will betrue or false. The originalstatement was neither .

B) Have students work in smallgroups or individually withteacher help at the board.

A) A basic idea necessary forunderstanding measurement shouldbe introduced. When we obtain ameasurement, we are attempting tomatch a discrete set of unitmeasures to a ctritinous thing.This.might be described in a broadsense as attempting to contsomething that is noncountable.

B) Have students work independtayor in small groups with teacherhelp.

A) By discussion and individualreading, students should solvethe problems in this unit. Somereteaching may be needed forindividuals.

B) Have students work togetherin small groups or individually.

Mm 21



A) AAvancina in Mathematicspp. 217-256

A) Advancing in Mathematicspp. /59-313

3) Hand rulers (inches)Ruler :inetric system)StringCardboard (flat and cylindrical)ProtractorYardstick


A) Pupils communicate math ideasby English sentences and learn thata group of symbols written in acertain order is a mach sentence.

Pupils realize that problemsinvolving math relationships mustbe read and analyzed carefully inorder to be translated into mathsentences

B) Administer suggested quiz atthe end of this unit or substituteteacher-made testing device.

A) Students should establish thatmeasurement is the counting of adiscrete set of unit measures.

Pupils exhibit a knowledge ofthe need for standards of measure.

Students should be able tomeasure perimeter of polygons, areasof regions, measures of angles,measures of circumferences, diameter:and radii of circles, and measuresof volume of rectangular prisms.


A) Advancing in Mathematics A) Students should increasepp. 3-45 computational skills by solving thevol. 3 teachers' manual four fundamental operations

using rational numbets.B) Administer suggested quiz at

the end of this unit or substituteteacher -made testing device

Mm 22

r 135

Concepts

IX. Symmetry and Congruence1. Balance in geometric figures2. Point symmetry investigations3. Sets of points4. Regular inscribed polygons5. Correspondence and congruence6. Congruent triangles8. Straightedge and ccmpass

X. Properties of Expressions1. Mathematical expressions2. Simplifying sum oe difference

forms3. Polynomials4. Multiplication in the system cf

expressions5. Proof- a reasonable explanation6. More about simplyifying

expressions7. Division in the ayetem of

expressions

XI. Equations and Inequalities1. Tools for problem solving2. Transforming equations3. Solving equations4. Solving inequalities5. The addition properties6. Simplifying to solve sentences7. English and mathematical

sentences8. Sentences with unknnwns on

both sides

136

A) Teaching MethodsB) Learning Acitvities

A) In this part we extend thestudent's experience with geometricrelations and geometric figuresby building on intuitive concepts.from earlier parts.

Background for this topic isprovided by investigation ofphysical objects that possess aparticular characteristic calledbalance.

t) Student should work with teacherin small groups or individually.

A) Expressions are important inmathematics because they are uue-ful in putting our thoughts intomathematical sentences. Equations

and inequalities often have theadditional advantage of makingthe structure of a situationeasier to see.

B) Have the students work insmall groups or individuallywith the teacher.

A) We should begin developingsystematic techniques for solvingequations and inequalities. In

the course of developing thesetechniques we acquire four newco,icepts.

In this unit we have occasionto make use of virtually :verythingwe know about the system of rationalnumbers.

B) Have students work in groupsor individually with teacher.

Mu) 23




A) Advancing in Mathematics A) Students relay tbe concept ofpp. 49-91 line symetry as it applies tovol. 3 teachers' manual physical objects and geometric

figures.B) Administer suggested quiz at

the end of this unit or substituteteacher-made testing levice.

A) Advancing in Mathematics A) Students are able to expresspp. 95-144 concepts by a mathematical expressionvol. 3 teachers' manual Students are provided practice

in determining the set of valuesan expression can have for a givenset of rational numbers.


A) Advancing in Mathematics A) Students should to able topp. 147-198 compute the results to problemsvol. 3 teachers' manual involving equations and inequalities,

Students develop an abilityto use English phrases and sentencesthat express mathematical ideas.


© 24

?'37

Co.:cepts

XII. Squares and Square Roots1. Pythagorean Triples2. Triples and triangles3. The Pythagorean Theorem4. Principle square root5. Intervals6. Determining aprrcximate

square roots7. Terminating and repeating

decimals8. Translating decimals into

fractions9. Closing gaps in the number

line

XIII. Similarity1. Scale drawings2. Similar figures3. Proving triangles to be

similar4. The tangent of an acute angle5. The sine and cosine of an

angle

138

A) Teachiog MethodsB) Learning Activities

A) An iAformal but meaningful-introduction to the system of realnumbers.

The first topic introduces theconcept of an ordered number triplewhose elements are positive integerrNext we introduce the concept ofa perfect square.

B) Work with students individuallyor in small groups.

A) The study of scale drawingsleads natJrally to the -onsiderationof geometric figures that have thesame shape but not the same size.Thus to the concept cf similarity.

The student leacns that twotriangles are similar if there isa correspondence between them suchthat the corresponding angles arecongruent and the ratios of themeasure of the correspondingsides are equal.

Then introduce the assumptionthat two triangles are similar iftwo angles of one are congruentto two angles of the other.

B) Wori: with students individuallyor in small groups.

H m 25



A) Advancing in Mathematicspp. 201-246vol, P teachers' manual


A) Students should realize thateach point on the number linehas exactly one real numbet co-ordinate and, conversely, thateach real number is the coordinateof a unique point on the numberline.


A) Advancing in Mathematics A) Students learn how to calculate

pp, 249-284 approximate values fog the tangent,

vol. 4 teachers' manual sine, and cosine of stch anglesby using measurements obtainedfrom carefully constructed diagams.

B) Administer suggested quiz atthe end of this unit or subst;.tuteteacher-made testing device.

M m 26

139

Concepts

XIV. Z,e Coordinate Plane1. Ordered Pairs2. Variables and ordered pairs3. Graphing sets of ordered pairs4. Graphing solution sets

140


A) Have students try to learnto think of matchingn of elementsof two sets in terms of orderedpairs.

Use graphs to represent orderedpairs and matching.

Also use ordered pairs innaming the solution sets ofequations in two unknowns.

Introduce variables and adependent variable. In orderto graph correspondences betweensets of numbers, introduce theconcept of a Cartesian coordinatesystem.

B) Work with students individuallyor in small groups.

M m 27


Resource:, C) PeopleEl) 'laces

A) Fxpected OutcomeEvaluation 8) Testing Program

A) Advancing in Mathematics A) Students show that a matchingpp, 287-313 can be thought of as a set ofvol. 4 teachers' manual ordered pairs.

Students show how raps canba used to represent sets ofordered pairs.

Students use the concepts ofindependent variable and dependentvariable of ao equation in studyingmathcoaticsi sentences thatdescribe problem situations.

B) Administer suggeuted quiz atthe erd of this unit or substitute.teacher-made testing doviee.

M m 28

141


CURRICULUM GUIDE

MATHEMATICS

HIGH SCHOOL

142

ALGEBRA I

SUGGESTED TIME ALLOTMENTS..1,ALGEBRA II /TRIGONOMETRY

) Symbols and Sets (6)

GEOMETRY

I) Sets of Numbers

and Axioms (10

2) Variables and

Open Sentences (9) 2) Open Seatr_nces in

One Variable (10)

3) Axioms, Equations

and Problem Solving3) Systems of Linear

(13)

) The Negative

Numbers (12)

5) Equations,Inequalities andProblem Solving

(10)

Open Sentences (12)

4; Polynomials and

Factoring (10)

) Rational Numbers

1). Lines and Angles (5)

) Inductive and

Deductive Reasoning(12)

) Constructions (5)

4) Triangles

Parallel and

6) Working with

Polynomials

(14)

7) Special Products

and Factoring (15)

) Working with

Fractions

(17)

Semester Exams

and Expressions (13)

6) Relations and

Functions (= 2)

7) Irrational Numbers

and Quadratic

Equations (15)

Perpe.ndicular Lines

(18)

6) ?olygons

7) Areas of

8) Quadratic Relations Polygons (13)

and Systems (15)

Semester Exams Semester Exams

M s

rSUGGESTED TIME ALLOTMENTS

ALGEBRA I ALGEBRA II/TRIGONOMETRY GEOMETRY

9) ExponentialFunctions andLogarithms (10)

8) Circles, Angles

and Arcs (10)

9) Graphs (16)

10) Progressions andflinomial Expansion

9) Measurement of

(10) Angles and Arcs (10)

10) Sentences in11) Polynomial

Two Variables (17) 10) Loc: (10)

Functions (10)

12) Matrices and 11) Proportion (7)

Determinants (10)

11) The Real Numbers

(17) 13) Permutatf.ons,Combinations and 12)

Probability

Similar Polygons(12)

(10)

13) Regular Polygons

12) Functions and 14) Trigonometric and the Circle (5)

Variation (10) Functions and14) Inequalities (6)

Complex Numbers(15)

13) Quadratic Equations

and Inequalities 15) Coordinate15) Trigonometric

U (15)Identies and

Geometry

(18)

Formulas (15)

14) Geometry and

Trigonometry (15) 16) The CircularFunctions andTheir Inverses

(15)_ 16) Trigonometry (7)

Review (5)

Semester Exams Semester Exams Semester Exams

144 s 2

PRE-ALGEBRA MATH

SUGGESTED TIME ALLOTMENTS

TRI NOM IRVANALYTICCFITIF.TIN

ADVANCED SENIOR MATHENATI S

1) Trigonometric 1) Sets 6 Numbers (5)

1) Numeration Functions of 2) The Algebra ofNumbers as a

Systems (15) Angles (15}1-

3) Extensions of the

(n

2) Operating with 2) Fundamental Logic of Algebra (15)

1-3 Base 7 and Relations (10)AD

Base 2 (20) T4) Inequalities,AbsoluteValues andCoordinate_ Systems(fl

3) Functions of

5) Functions and TheirTwo Angles (15) Graphical

Representation (3)3) Divisibility,

Factoring, andSets (10)

) Liaear and

4) Solution of Quadratic4) Whole Numbnrq

Equat!_ons 'JO)

and Operations (10) Triangles (15)

7) Determinants (10)

5) Rational and 5) Vectors (5)

Irrational Nup')ers6) Line Values of ) Polynomial

0 of Arithmetic (15)

the Trig. Gradha Functions (12)(10)

7) Polar Coordinates6) Percent and (10) 9) Permu:ations,

Its Uses (20) Combinations and the

8-9) LogarithTz (10) Binomial Expansion(13)

Semester Exams Semester Exams Semester Exams

145

PLE-ALGEBRA MATH

SUGGESTED TIME ALLOTMENTS

GimmyyRvIklaVANCED SENIOR MATHEMATIC1TRIGONOMETRY/ANALYTIC

Aa

7) The Real Numbn's

of Arithmetic (15)

10) Functions (5) 10)

11)

Analytic Preparation

(8)11) Coordinatesand Graphs ( )

12)

13)

The Straight Line(10 Plane Analytic

Gemetry (13)8) Directed Numbers -

Addition and

Subtraction (20) Conic Sections (15)

12) Functions (9)

9) Directed Numbers

Multiplication

and

14) Curve Tracing (5)

13)

14)

Limits (8)

15-16) Polar Coordinates

and Locus Problems

(5)Derivatives

09)-3tg

Division (20)

17) The Derivative (10

of Algebraic

Functions (15)

10) Basic Geometric

Figures (15)

18) Differentiation

of Algebraic

Functions (10)

15) Applications (10)

(12)16) Integration

- 19) App: ications of

the derivative (10)11) Areas and Volumes

(10)

20) Integration (1012) Line and

Circle Constructions

CO)

17) Applications of the

Definite Integral(15)

Semester Fxams Semester Exams Semester Exams

146 M a o

A) Teaching Methods


Pre-Algebra Math

I. Numeration Systems-The decimal numeration system A)

-Place values of basic numerals-Egyptian numeration system-Roman numeration system- Positional system-Exponential notation-An abacus and the base-tensystem

-Abacus for base seven-Adding and subtracting, base 6)

seven-From base ten to bass seven

The teaching methods used in pre-algebra math include:1) Lecture and demonstration.2) Involving the student through

various projects which relateto the unit being studied.

3) Working in small groups toassist those who are havingdifficulty.

Student activities include:1) Working assigned problems.2) Keeping a homework notebook.3) Projects which illustrate the

concepts of the lesson.

II. Base Seven and Base Two-Multiplication tables A)

-Base seven multiplication table &) Refer :o chapter' 1-Division, base seven B)

-Multiplication table and divi-sion

-The binary numeration system-Addition and subtraction,binary numeration system-Multiplication, binary system-How many digits in a numeral?- Positional numeration systems

III. Divisibility, Factoring and Sets- Divisibility of whole numbers-Divisors of whole numbers- Factors and prir'.e numbers

-Complete factorization- Sets

-Finite and infinite sets- The e4ty set-Operations with sets-Greatest common factor-Least common multiple

A)

&) Refer to chapter 1B)

M s 5

147

A) irintedB) Audio Visual


A) Twnt: Pret-llabraaittituaU21NicholsHolt. Rinehart & Winston.

Text: Chapter 1

A) Text: Chapter 2

A) Texts Chapter 3

A) Expected OutcomeE-aluatiol B) Testing Program

A) Students are expected to demon,- .

strate a functional understandingof numeration systems whetherdecimal or othor.

B) Teacher-made test

A) Students should demonstrate theability to perform the funda-mental operations in differentnumber bases.


A) Students should exhibit theability to distinguish b.tweenprime and composite numbers,to factor composite numbers,and to find the lowest commonmultiple and the grsatest com-mon factor.Teacher -wade test

148H s 6

Al Teaching Methods

Concepts B) Learning A:tivitles

IV. Whole Numbers and Operations-The number zero A)

-The number one &) Refer to chapter 1

-The commutative properties B)

-The associative properties-The left distributive property-A mat%ematical deduction orderivation

-Closure-Addition and subtraction asinverse operations

-Multiplication and divisionas inverse operationo

- How to solve problems

V. Rational and Irrational Numbersof Arithmetic

- Natural numbers and eationel A)

numbers of arithmetic &) Refer to chapter 1-The arithmetic mean B)-Numbers between two givennumbere

-Comparing rational numbers ofarithmetic

-Reciprocals-TerrAnatis,g decimal numerals-Non-terminating repeatingdecimal numerals

-Distributive properties-Non-repeating, non-terminatingdecimal numerals

VI. Per lent and Its Uses-What :loos %(P er cent) mean's A)

-Per cents and decimal names &) Refer to chapter 1-Per cent and rational numbers B)

-nett-)

-Ratio and per cent-Proportion-Discount-Commission-Per cents less than one-Per cent of change

149Ms7


Resources C) Peoplt,D) Places

A) Text: Chapter 4

A) Text: Chapter 5

A) Texts Chapter 6


A) Students Should demonstrate afunctional knowledge of thecommutative, associative,distributive, and closureproperties of the number system.


A) Students should demonstrate afunctional knowledge of thedifference between rational andirrational numbers, and how tocompute arithmetic means.

B) Teacher -made test

A) Students should exhibit computa-tional skills in solving elltypes of per cent problems.

B) Teacher-made teat

(-0

M s 8

A) Teaching Methods


VII. Real Numbers of Arithmetic-Square root of a number A).,Real numbers of arithmetic &) Refer to chapter 1-Pythagorean triples B)

-The property a2 4-b2 = c2

-Rational numbers and the ray-Ray and irrational numbers-The complement of a setOne-to -one correspondence-One-to-one correspondencebetween two infinite acts

VIII. Directed Numbers - Additionand Subtraction-Number line and one-to-one A)

correspondence &) Refer to chapter 1-Integers B)

-Directed real numbers-Uses of directed numbers-Adding directed numbers-Absolute value-Coumutativity of addition-Associativity of addition-Relation between addition andsubtraction of directed numbers

-Closure of sets of directednumbers

IX. Directed Numbers - Multiplicationand Division-Properties of multiplication A)

of directed numbers &) Refer to chapter 1-The left and right distributive B)

properties- Multiplication; ,and division of

directed numbers-Reciprocals and division-Dividing by uultirlying

M s 9

151


Resources C) PeopleD) Placas

A) Text: Chapter 7

A) Text: Chapter 8

A) Text: Chapter 9

152


A) Students should demonstratecomprehension of the basicelements of set theory andthe ability to compute squareroots.


A) Students should exhibit computa-tional skills in adding andsubtracting of signed numbers.


A) Students should demonstrate th .ability to multiply and dividesigned numbers accurately.

B) Teachbr-made test

s 10

A) Teaching Methods


X. Basic Gsometric Figures-Points and segments A)

-Polygons &) Refer to chapter 1- Half-lines and rays B)

-Angles-Measure of segments and angles- Classification of triangles-Interiors of angles- SiCIple closed curves

-Building goometric objects outbf points

-Interiors of triangles

XI. Areas and Volumes-Rectangles A)

-Triangles &) Refer to chapter 1

-Parallelograms B)

-Regular polygons- Equilateral triangles- Regular polygons and circles-Inscribed hexagons-Inscribed regular polygons-Prisms-Cylinders-Pyramids-Cones

XII. Line and Circle Constructions-Ralf-lines. rays and angles A)

-Midpoints &) Refer to chapter 1- Constructing angles B)

-Angle bisector-Inscribed angles-Ratio and geometry-Similar polygons-.Similar triangles-Ratios in similar triangles-Angles inscribed in semi-Urcles

- Constructing line segments-Distance from a point to aline

- Perpendiculars-Central angles

53M s 11


Resoumeb G) PeopleD) Places

A) Text: Chapter 30

A) Text: C4apter 11

A) Text: Chapter 12

154


A) Students should demonstrate aworking knowledge of the basicprinciples of Euclidian geometry.


A) Students 'should exhibit skillsin computing areas and volumesof the more common geometricfigures.


A) Students should demonstrateproficiency in constructing theelementary geometric figures,and a functional knowledge ofratioa in triangles.

B) Teacher -made test

M s ]2

Concepts

Algebra I

I. Symbols and Sets .

Numbers and Their Relationships-Representing numbers on a line:Order relations-Comparing numbers:the sign ofequality

- Comparing numbers: the signs, Ofinequality

Grouping Numbers in Sets and Subsets- Meaning of membership in a set-Kinds of sets- The graph of a set-How subsets relate to sets

Using Numbers in One or MoreOperations

- Punctuation marks in algebra-Order of operations

II. Variables and Open SentencesAnalyzing Algebraic Statements-Evaluating algebraic expressionscontaining variables

- Identifying factors, coefficients,and exponents

- Solving open sentencesProblems Solved With Variables- Thinking with variable3: fromsymbols to words

-Thinking with variables: fromwords to symbols-Solving problems with opensentences


A) The basic teaching methods usedin Algebra I are:1) teacher explanation and

demonstration;2) students working at the board

with individual help;3) ure of various audio-visual

aids such as: films, film-

strips, transparencies, andthe chalkboard.

B) The suggested learning activitiesare:1) students working assigned

problems at their desks with2) keeping a notebook of class

totes and assign rents;3) individual work from the

programmed supplement pro-vided with the text;

4) have students help each otherwith problems.

Refer to the teachers? manual, pp.6-8 for more suggested teachingmethods.For suggested student assignmentsrefer to the teachersg manual pp.48

A) Use methods 1-3 as indicated inchapter 1.Refer to the teachers" manualpp. 8-10.

B) As indicated in chapter 1.For suggested assignments, referto teachers' manual, pp. 48-49.

M s 13



A) Text: ModapAlgerAAStructureand Method

Programmed practice to accompanythe above text.

B) Filmstrips, transparencies asavailable

Chalkboard, various math modelsC) Teacher, other members of the

department

A) Texts Chapter 2

156


A) Students demonstrate comprehensionthat:1) the number line order:; the

numbers of arithmetic;2) a set is determined by its

elements;3) the site or complexity of a

gumecal does Da indicatethe size of the numberrepresented;

4) in a one-to-one correspondence,each element of a set ispaired with an element of theother set, and that nu elementis omitted or repeated.

B) Administer progress and/or teacher-made tests and quizzes as required.Observation of students.

A) Students should exhibit under-standing of:1) the steps involved in the

algebraic solution ofproblems;

2) the concept of a variable;3) the basic concept of "factor".

B) Use provided progress test and /orteacher-made test.

Observation.

M s 14

Concepts

III. Axioms, Equat-ms, and ProblemSolvingIdentifying and Using NumberAxioms

-Axioms of equality-The closure properties- Commutative and associativeproperties of arithmeticnumbers

-The distributive property;special properties of 1 and 0

Transforming Equations WithEquality Properties

-Addition and subtractionproperties of equality

- Division and multiplicationproperties of equality

-Combining terms and usingtransformation principles

-Equations having the variablein both members

IV. The Negative NumbersExtending the Number Line

-Directed numbers- Comparing numbers

Operating With Directed Numbers-Addition on the number line-The opposite of a directednumber

-Absolute value-Adding directed numbers-Subtracting directed numbers-Multiplying directed numbers-Dividing directed numbers-Averages and directedrr.mbers (optional)

A) Teaching MethodsB) Learnirg Activities

A) Refer to ,:hapter 1 for methods.B) The learning activities are as

indicated in chapter 1.For further methods and suggestedassignments, refer to the teachers:manual, pp. 10-13 and 49-50.

A) Refer to chapter 1.See also teachers? manual pp. 13-17.

B) Refer to dapter 1.For assignments, see teachers?manual, pp. 50-51.

15714 s 15


Resources C) PeopleD) ;laces

A) Text: Chapter 3

A) Text: C)apter 4

1.'58


A) Students should manifest aclear understanding of:1) the formal axioms and

postulates of algebra;2) the necessity for clear

and concise mathematicallanguage;

3) the basic methods employedin formal proofs.

B) Use progress test and/orteacher -made test.

Observation.

A) Students should demonstratethe ability to perform thebasic operations with thedirected numbers.

B) Progres6 tests and/or teacher -made tests

Observation

M s 16

Concepts

V. Equations. Inequalities, andProblem SolvingOpen Sentences in the Set ofDirected Numbers

-Transforming equations-The properties of inequality-Pairs of inequalities (optional)

The Analysis of Problems-A plan for solving problems-Problems about consecutiveintegers

-Problems about angles- Uniform motion problems- Mixture problems

VI. Working With PolynomialsAddition and Subtraction ofPolynomials- Adding polynomials- Subtracting polynomials

Multiplication of Polynomials-The product of powers- The power of a product-Multiplying a polynomial by amonomial

-Multiplying two polynomials- Problems about areas- Powers of polynomials

Division of Polynomials- The quotient of powers- Zero as an exponent(optional)- Dividing a polynomial by amonomial-Dividing a polynomial by apolynomial

159


A) Refer to chapter 1, and to theteachers° manual pp. 18-22.

B) See chapter 1.For assignments, see teachers°manual pp. 51-52.

A) Refer to chapter 1 and to pp. 22 -25 of the teachers° manual.

B) Learning activities as indicatedfor chapter 1.

For suggested assignments seePP. 52-54 of the teachers' manual.

M s 17



A) Text: Chapter 5

A) Text: Chapter 6

166


A) Students should display:1) understanding of the

transitive, additive, andmultiplicative propertiesof Lieouality;

2) the ability to solve linearequations;

3) tie ability to set up andsolve problems.

B) Duc, to the inherent difficultyof this material, testLg shouldcover smaller blocks of materialin the form of quizzes. I.chapter test may be used also.

Observation.

A) The students e.hould demonstratecompetence Ail the fundamentaloperations on algebraicpolynomials.

B) In this area also, quizzesshould be given coveringeach of the areas of com-petence in dealing withpolynomials.The progress tests may alsobe used.

Obs,3rvation.

N s 18

Concepts

VII. Special Products and FactoringThe Distributive Property inFactorirc-Factoring in algebra-Identifying common factors- Multiplying the sum anddifference of two numbers-Factoring the difference oftwo squares

Quadratic Trinomials- Squaring a binoralal: plateausection

-Factoring a trinomial square-Multiplying binomials at sight-Factoring the product ofbinomial sums or differences-Factoring the product of abinomial sum and a binomialdifference-General method of factoringquadratic trinomialc

Extension of Factoring-Combining several types offactoring-Working with factors whoseproduct is zero-Solving polynomial equationsby factoring-Using factoring iqfproblemsolving


A) Refer to general methods outlinedin chapter 1, and to pp. 25-2Rof the teachers' manual.

B) As indicated in chapter 1.Suggested assignments are givenon pp. 54-55 of the teachersmanual.

14 s 19

C

A) PrintedB) Audio Visual A) Expected Outcome

Resources C) People Evaluation B) Testing Program

D) Placers

A) Text: Chapter 7

162

A) The students should displayan understanding of thevarious types of factoringas evidences by theirability to successfullyfactor these types.

B) Short quizzes, teachermadeand progress tests should .

comprise the testing this

section.Observation.

II s 20

Concepts

VIII. Working With FractionsFractions and Ratios- Defining algebraic fractions-Reducing fractions-Ratio- Per cent and percentage pro-blems

Multiplying and DividingFractions-Multiplying fractions-Dividing fractions- Fractions involving multipli-cation and division

Adding and SubtractingFractions-Combining fractions withequal daaominators

-Adding fractions with unequaldenominators

- Mixed expressions-Complex fractions (optional)Fractions in Open Sentencesand Problems-Open sentences with fraction-al coefficients

- Investment problems- Per cent mixture problems- Fractional equations-Work problems-Motion problems

63


A) Refer to methods indicated inchapter 1 and to pp. 28-31 ofthe teaches manual.

B) See chapter 1 for learningactivities. Suggested assign-ments are outlined on pp. 55-57in the teachers9 manual.

M s 21


resources C) PeopleLi) Places

A) Text: Chapter 8

164

A) Expected OutcomeEvaluatio...1 B) Testing Program

A) The students should exhibit aclear understanding ofalgebraic fractions. Theyshould be able to performthe basic operations withfractions with a highdegree of accuracy.

B) Use progress tests to supple-ment teacher-made tests andquizzes.

Observation.

M s 22

Concepts

IX. GraphsOrdered Pairs of li,mbers andPointe in a Plane-Open sentences in twa variables-Coordinates in a plane

Linear aivations and StraightLines-The graph of a linear equationin two variables

-Slope of a line- The slope-intercept form of alinear equation-Determining the equation of aline

Inuqualities and Special Graphs-Graph of an inequality intwo variables-Graphs that are parabolas

Statistical Graphs-Breen-line and bar graphs-Circle graphs

X. Sentences in Two VariablesSolving Systeas of Linear OpenSentences

- The graphic method-The addition and subtractionmethod-Problems with two variables- Multiplication in the additionand subtraction method-The substitution method-Graphs of pairs of linearinequalities (optional)

Additional Problems-Digit problems-Motion problems-Age problems-Problems about fractions


A) Methods as outlined in chapter 1.Further suggestions on pp, 31-33of teachers° manual.

B) As indicated for chapter 3..Suggested assignments on pp. 57-59 of teachers/ manual.

A) See chapter 1 for teachixgmethods.More suggestions may be foundon pp. 33-35 in the teachers°manual.

M s 23



A) Text: Chapter 9B) Graph board

A) Text: Chapter 10B) Graph board

166


A) Students should demonstratethe ability.to:1) represent points, lines,

and regions graphically;2) graphically solve linear

equations and inequalitiesin two Ilnknowns;

3) determine the slope of aline and the equation,given the slope and the"y" intercept.

B) Progress tests and teacher -made testsObservation

A) Students should manifest:1) an understanding of

simultaneous linearequations in two un-knowns;

2) the ability to solvetwo equations in twounknowns by graphing,adding and subtracting,and substitution.

B) Progress tests and teacher-7made testsObservation

M s 21,

Concepts

XI. The Real NUmbersThe System of Rational Numbers

-The nature of rational numbers-Decimal forms of rationalnumbers

Irrational Numbers-Roots of numbers-Properties of irrationalnumbers-Geometric interpretation ofsquare roots

Radical Expressions-Multiplication, division, andsimplification of radicals

-Addition and subtraction ofradicals-Multiplication of binomialscontaining radicals

-Radical equations

XII. Functions and VariationSelecting Pairs of Numbers-Relations-FunctionsVariation-Direct variation and proportion-Inverse variation-Joint variation and combinedvariation (optional)

XITT. Quadratic Equatior. and Inequa-litiesGeneral Methods of SolvingQuadratic Equations-The square-root property-Checking solution sets-Completing a trinomial square-The quadratic formula-The nature of the roots of aquadratic equation (optional)

The Solution of QuadraticInequalities-Solving quadratic Inequalities-Using graphs of equations tosolve incqualities (optional)


A) See chapter 1 for suggestedmethods.Aso refer to teachers' manual,

PP. 35-37.B) Suggested learning activities

outlined in chapter 1.Possible assignments can be foundon pp. 60-62 of teachers' manual.

A) As indicated in chapter 1.See also the teachers/ manual,pp. 37-40.

B) See chapter 1.Suggested assignments, pp. 62-63in teachers' manual.

A) See chapter 1 and pp. 40-43 ofteachers' manual.

B)lautlined in chapter 1.SuggPstci assignments on pp. 63-64 of teachers' mannal.

I 6M s 25


Resources C) PevleD) Places

A) Text: Chapter 11

A) Text: Chapter 12

A) Text: Chapter 13

168


A) Students should exhibit:1) a functional understanding

of the system of rationalnumbers and irrationalnumbers;

2) the ability to simplifythe various norm;, ofradical expressions;

3) a basic knowledge ofradical equations inclu-ding the solution thereof.

B) Use progress tests and teacher-made tests as needed.

Observation.

I) Students should demonstrate:1) the ability to differentiate

between relation and function;2) an understanding of the

types of variation.B) Teacher-made test and progress

testsObservation

A) The students should exhibit:1) the ability to solve

quadratic equations bythe various methods;

2) an understanding ofquadratic inequalities.

B) Progress tests and teacher-made testaObservation

M s 26

Concepts

Geometry

I. Lines and Angles-Undefined terms-Definitions-Gcnstructions

II. Indiotive and Deductive Reasoning-Discovering information byobservation, measurement andexperiment

-Distinguishing between induc-tive and deductive reasoning

-Basic assumptions- Proof in geometry-The formal proof

III. Constructionsthe formal procedure in ageometric construction

- The nine basic constructions


A) The teaching mAhods in geometryinclude:1) Teacher lecture, demonstratioi-'2) Proving theorems at the black-

board3) Using audio visual aids as

available.B) The learning activities of the

students include:1) Working assigned problems at

desks with teacher assistance2) Keeping a notebook of class

notes and assigned problems3) Working in teams at the board

A)


A)

&) Refer chapter 1B)

M s 27



A) Text: pilmjImoIrxWeichons, Krickenberger,and Pearson

Ginn and Company.Text: Chapter 1

A) Text: Chapter 2 & 3

A) Text: Chapter 6


A) Students should demonstratethat they know the definitionscovered in the chapter.

B) Quizzes and teacher-made tests

A) Students should illustrate:1) that they can distinguish

between deiuctive and in-ductive reasoning;

2) a working usage of thebasic assumptions;

3) the ability to set up aformal proof.


A) Students should be able toperform the basic geometricconstruction.

B) Teacher-made constructions test

170M s,28

Concepts

IV. Triangles-The basic types of triangles-The concept of congruence-Proving triangles congruentby side-angle-side and angle-side-angle

-Prove theorems 1-5

V. Parallel and Perpendicular Lines-Theorems 6-16-Converses, inverses, andcontrapositives

-Symmetry-Planes and introductory solidgeometry

VI.. Polygons

-Theorems 17-29-Applications of theorems topractical problems

VII. Areas of Polygons-Theorems 30*37-Radicals and qutdraticequations

-Finding the areas of irregularpolygons

-,volvmea of geometric solids


L)&) Refer to chapter 1B)

A) Refer to chapter IUse solid models to demonstrateand aid introduction to solidgeometry

B) Refer to chapter 1

A)


A)


171M s 29

t.



A) Text: Chapter 4

A) Text: Chapter 5

A) Text: Chapter 7

A) Text: Chapter 8

172


A) Students should show:1) the ability to distinguish

among the types of triangles;2) an understanding of congruence;3) the ability to prove triangles

ccngruert;4) that they can prove theorems

formally.B) Teacher-made tests

A) Students should demonstrate amastery of:1) the -,:roof of the formal

theorems;2) the converse, inverse and

contrapositive of a theorem;3) the various types of symmetry,

B) Teacher-made tests

A) Students should exhibit theability to:1) distinguish among the types

of polygons;2) prove the formal theorems

related;3) apply the concepts in the

theorems to problem solving.B) Teacher made tests and quizzes

A) Students should be able to:1) prove the theorems in the

chapter;2) apply the concepts to problem

solvIng;3) find the areas of irregular

polygons by the trapezoidAm6bod.

B) Teacher-made tests and quizzes

M s 30

A) Teaching Methods

Concepts D) Learning Activities

VIII. Circles, Angles, and Arcs- Assumptions relating to circles A)

-Theorems 38-47 &) Refer to chapter 1

-Solid geometry B)

a) right circular cylinderb) right circular conec) sphere

IX. Measurement of Angles and Arcs-Theorems 48-52 A)

-Numerical relations between &) Refer to chapter 1the arcs of a circle and the B)

angles associated with acircle

X. Loci-Locus of points A)

- Introduction to the basic &) Refer to chapter 1conic sections B)

-Theorems 53-60

XI. Proportion and ProportionalLine Segments- Basic theorems'of proportion- Proportion as related totriangles-Theorems 61-64

A)


17 3M s 31


Resources O) PespleD) Places

A) Text: Chapter 9

A) Text: Chapter 10

A) Text: Chapter 11

A) Text: Chapter 12

174

A) Expected OutcomeEvaluation B) TeAing Program

A) Students Lhould show the abilityto:

1) prove the theorems and applythe concepts;

2) recognize and work .

with the concepts related tothe solid geometry figuresdiscussed.


A) Students should demonstrate thability to:1) prove and apply the form

theorems;2) solve proble.as related to 11

arcs and angles of a circleB) Short quizzes are best in thi:,

chapter.

A) The students should demonstratethe ability to:1) determine a locus from a s,

of conditions;2) prove the locus theorems;3) recognize the basic conic

tions from their locus delitions.

B) Teacher -made tests

A) Students should demonstrate aa workinf, knowledge of:1) the basic proportion propel-2) the proportional division 01

triangles.B) Teacher-made test

s )2

Concepts

XII. Similar Polygons-Theorems 65-78-Using similar polygons inproving line segments equalor proportional

XIII. Regular Polygons and the Circle(Optional)

-A brief overview of the rela-tions between the circle andpolygons

XIV. inequalities-Relations between unequal linesegments and angles

-Theorems 89-101

XV. Coordinate Geometry-Using algebraic methods ingeometry-The straight line-Distance between two points-Graphs of linear equations-The conic sections: circle

parabolaellipsehyperbola

XVI. Elements of Trigonometry-The basic trigonometricfunctions

-Solving triangles

175


A)


A)&) Informal discussion and lecturesB)

A)


A) Due to the inherent difficulty inthis area, the teacher shouldtake care to explain thoroughlythe concepts outlined. This areaShould be emphasized as it isvery important.

B) Board work and working in pairson the concepts.

A) Refer to chapter 1B) Refer to chapter 1

Practical experience in determiningdistances by indirect measure-ment. Construct basic transitsfor indirect measurement.

M s 33

A) PrinteCB) Audio Visual


A) Text: Chapter 13

A) Text: Ch_pter 14

A) Text: Chapter 16

A) Text: Chapter 17Anal is Geometryn Introductimito Calculus, Schock & WarshawPxentice-HallChapters 3 & 4(Available from department Chair-man)

A) Text: Chapter 15B) Student constructed transits

"76

A) Expected Out3omeEvaluation B) Testing Program

A) Students should manifest afunctioning comprehension ofthe interrelation of similarpolygons and proportional lines.


A) The students should demonstratea familiarity with the conceptsdiscussed.

B) None required

A) Studert3 should show the abilityto work with inequalities.


A) Students should demonstrate aworking knowledge of the variousaspects of the concepts outlined.


A) Students should show the abilityto solve triangles usingtrigonometric functions.

B) Teacher-made testsIracticals

M s 34

Concepts

Algebra II and Trigonometry

I. Sets of Numbers; AxiomsSets in Algebra- Sets and their relationships-Sets and variables

Axioms for Real Numbers- Order in the set of real numbers-Properties of addition andmultiplication- Identity and inverse elements

Using Axioms-The concept of Proof-Adding and Subtracting realnumbers-Multiplying and dividing realnumbers

II. Open Sentences in One VariableEquivalent Expressions-Algebraic expressions-Simplifying expressionsEquivalent Open Sentences- Equivalent equations-Equivalent inequalities-Combining open sentences-Statements end conversee4sing Variables to Solve Problems-Interpreting algebraic expressionsand sentences- Solving problems

III. Systems of Linear Open SentencesLinear Equations and their Graphs-013:41 sentences in two variables- Graphs of lircAt equations intwo variablesLines and their Equations-The slope of a line-Determining an equation of a line-Systems of tvo linear equationsin two variables

Linear InerualWes


A) The basic teaching methods usedin Algebra II/ Trig. are:1. Teacher explanation anddemonstration2. Students !corking at theboard with individual elp3. Use of Audio Visualaids such as filmstrips, films,math models, transparencies,and the chalkboard.

B) Learning activities include:1. Students working assignedproblems at desks with helpavailable.2. Keeping a notebook of classnotes and homework.3. Having better students assistthose students with problems.4. Students working at theboard.

The teachers' manual has more detailedsuggestions in this area.

A)

&) Refer to chapter 1 andD) teachers' manual

A)

&) Refer to chapter 1 andB) tesch...2rs' manual

17 7Ma 35




A) Text: Modern Algebra and A) Students should show abilityTrigonometry to Work with field,.axioms'andHoughton Mifflin .order relations in the realchapter 1 number system.

B) Teacher-wade and/or piogregeB) Film: Introduction to Irrational tests

Numbers F.O.M.

A) Text: chapter 2

B) Film: introduction to FunctionsF.O.M. # 1126

A) Text: chapter 3

178

A) Student should exhibit skills inin solving linear equationsand inequalities in one variable.

B) Progress test and/or teacher-madetests

A) Students should demonstrate theability to solve two equationsin two unknowns by the usualmethods.

B) Teacher made and/or progresstests

M s 36


IV. Polynomials and FactoringProducts of Polynomials A)-Positive integral exponents &) Refer to chapter 1 and-Multiplying polyonomials B) teachers' manualFactoring- Factoring monomials-Factoring polynomials-Factoring quadratic trinomials-Greatast common factors andleast common multiples of poly-nomialsFactoring in Open Sentences-Using factoring to solve equations-Using factoring to solveinequalitiesDivision of Polynomials-Dividing polynomials- The Fac'or Theorem (optional)

V. Rational Numbers and ExpressionsExtension of Laws of exponents A)-Zero and negative exponents &) Refer to chapter 1 and

Operations with fractions B) teachers' manual-Rational algebraic expressions- Simplifying rational expressions-Multiplication and division-Addition and subtraction-Complex fractions

Using Fractions to solve Problems-Fractional coefficients- Problems involving fractions-Fractional equations

Decimals; Scientific Notation-Decimals for rational numbers-Approximations

M s 37


Resources C) PeopleD) Placrs

A) Text: chapter 4

A) Text: chapter 5

u

180


A) Students should exhibit a masteryof the formal techniques of cal-culating with polynomials andthe standard factoring procedures

B) Progress test and teacher madetests.

A) Students should demonstrate theability to perform the operationwith rational numbers andrational algebraic expressions.

B) Progress tests and teacher madetests.

1,4 a 38


VI. Relations and FunctionsSpecifying Relations and Functions- Relations

-FunctionsUnear Functions and Relations- Linear functions and direct

variation-Special functions and relations(optional)Quadratic Functions & Relations-Quadratic functions and variation- Quadrgic functions specifiedby y=ax + bx + c- Quadratic unctions specifiedby y=a(x-h) + k-Using graphs of quadratic functions

VII. Irrational Numbers and QuadraticEquationsReal Roots of Real Numbers- Using radicals to express roots

-R.tional and irrational roots-Rational operations-Decimals for real numbersWorking with Radicals-Properties of radicals-Simplifying sums of radicals-Products of sums containingradicalsRadicals in Equations-Using radicals to solvequadratic equations- Relations between roots andcoefficients of a quadraticequation-The nature of the roots of aquadratic equation-Solving quadratic inequalities-Irrational equations

A)&) Refer to chapter 1 and

B) teachers' manual

A)&) Refer to chapter 1 andB) teachers' manual

181 s 39

tA) PrintedB) Audio Visual


A) Text: chapter 6

B) Film: Quadratic Equations and theirtheir Solution.,F.O.M. 111114

Introduction to FunctionsF.O.M. #1126

A) Text: chapter 7

182


A) Students should illustrate thesuccessful E asp of the conceptof function.

B) Progress tests and teacher madetest.

A) Students should be able toperform t11e operations withradicals and the solution ofquadratics.

B) Progress tests and teacher madetests

M s 40

A) Teaching Methods


VIII. Quadratic Relations andSystemsCoordinates and Distance in aPlane

-Distance between points- Perpendicular lines(optional)Graphing Quadratic Relations

-Circles- Parabolas

- Ellipses

-Hyperbolas-Inverse variationsSolving Quadratic Systems

-Graphic solutions- Linear quadratic systems:substitution- Quadratic-quadratic systems

A)&)Refer to chapter 1 andB) teachers' amnual

X. Progressions and Binomial ExpansionsArithmetic Progressions A)- Arithmetic progressions .;)Refer to chapter 1 and

-Arithmetic means B) teachers' manual- Sum of an arithmetic

progressionGeometric Progressions and Series-Geometric pror'ression-Geometric means-Geometric series- infinite geometric series

Binomial expansions-Powers of binomials- The general binomial expansion

IX. Exponential Functions and LogarithmsFrom Exponents to Logarithms A)-Rational numbers as exponents &) Refer to chapter 1 and-Real numbers as exponents B) teachers' manual- Exponential and Logarithmic functionsUsing Logarithms-Common logarithrInterpolation-Products and quotients-Powers and roots- Comb!.ned operations

-Using logarithms to solve M s 41equations

183



A) Text: chapter 8

B) Film: The ParabolaF.O.M.

Model of a cone (dissectable)

A) Text: chapter 13

B) Film: Arithmetic SeriesF.O.M.

A) Text: chapter 9

B) Film: Exponents and LogarithmnF.O.M. #1117

1 4


A) Students should demonstratecomprehension in the basicmethods of analytic geometry.

B) Teacher made and progress tests

A) Students should demonstrate amastery of arithmetic andgeometric series.

B) Teacher made and progress tests.

A) Students should exhibit the Rbilit;to compute and use logarithmsin computation.

R) Teacher made and progress tests.

M d 42

Concepts

XI. Polynomial FunctionsPolynomial Functions over theComplex Numbers- Synthetic substitution-Remainder and factor theorems-Depressed equations-The fundamental theorem ofAlgebraPolynomials with Real Coefficients-Descartes' Rule-Graphing- Location Principle- Upper and lower bounds for roots- Polynomials and linear interpola-

tion

XII. Matrices and DeterminantsMatrices- Terminology-Addition and scalar multiplication-Matrix multiplication- The determinant function- 2x2 Matrices

-The inversa of a matrix-Solution of systems of linearequationsDeterminants- Expansion by minors-Properties of determinants-Cramsar's Rule


A)&Maier to chapter 1 andB) teachers' manual

A)&)Refer to chapter 1 andB) teachers' manual

XIII. Permutations, Combinationsand Probability

Permutations A)-Fundamental counting principles &) Refer to chapter 1 and-Linear and circular permutation B) teachers' manual- Permutations of elements the samCombinations-Counting subsets- 'Those formed from several sets-Binomial Theorem and Pascal'sTriangleProbability-Sample spaces and events- Melning of math probability-Mutually exclusive events 21 s 43

-Independent at,d dependentevents 185


Resources C) People0) Plai-as

A) Text: chapter 14

A) Text: chapter 15

A) Text: chapter 16

181


A) The students should have aworking knowledge of the algebraof real numbers and complexnumbers

B) Progress tests

A) Students should exhibit a compu-tational ability in solvingdeterminents.

B) Teacher made tests.

A) Students should demonstrate amastery of the elementary notionsused in counting and inprobability.

B) Progress tests.

14 s 44

A) Teaching Methods


XIV. Trigonometric Function:; andComplex numbersCoordinates and Trigonometry-Rays and angles and points-Sine and cosine functions- The trigonometric functions-Special anglesEvaluation rind applying trigono-

metric functions-Using tables- logarithms of the values of

Trigonometric functons- Reference angles

Vectors-Adding vectors

- Resolving vectorsWorking with Complex Numbers

- Complex Numbers-Multiplying pure imaginary numbers-Complex numbers and quadraticequations

XV. Trigonometric Identitiesand FormulasIdentities and One Angle-The Fundamental identities-Proving identitiesIdentities Involving Two Angles

-The distance formula-The cosine of the difference oftwo angles-Functions of sums and differencesof angles-Double and Half angle identities-Sum and product identitiesTriangle applications-The Law of cosines- The Law of Sino.s

- Solving triangles-Areas of Triangles

181

A)

&) Refer to chapter 1 andB) teachers' manual

A)&) Refer to chapter 1 andB) teachers' manual

M345



J.) Text: chapter 10

B) Film: Law of CosinesF.O.M. #1125

A) Text: chapter 11

188


A) Students should demonstrate aworking knowledge of the basictrignome'xic functions.

B) Progress tests

A) The students should exhibit theability to work with the basictrig. identities.

B) Progress tests

14 s 46


XVI. The Circular Functions and theirInversesVariation and Graphs A)

- Measuring. arcs and angles ReEer to chapter 1 and-The circular functions B) teachers' annual- Graphs of Cosine ans Sine funetionS- Graphs of other circular functionsInverse Functions and graphs-Inverse va?ues-Inverse circular functionsC,)en Sentences-General and particularsolutions-Additional sentences

I891 m s 47



A) Text: chapter 12

A) Expected OutcomeEvaluation B)Testing Program

A) The students should manifest aclear understanding of thegraphs of the trig. functions.

B) Progress tests

Ms 48

Concepts

Trigonometry/Analytic geometry

I. Trigonometric Filnctions of Angles


A) The teaching methods for trigonometry do not vary noticeeMy fromother mathematics disciplines,i:e.1) lecture;2) etudents working assigned

problems at their seats;3) discussion;4) individual help as required.

B) Suggested learning activitiesinclude:1) students keep a notebook of

class notes and assignedexercises;

2) working problems at the board;3) students helping each other to

better understand.

Fundamental Relations A)&) Refer to chapter 1B)

III. Functions of Two Angles A)


IV. Solution of Triangles A)


B) Have students construct simpleline of sight transits.

191 s 49



A) Text: Modern Tri.onomelaWelchans;' Krickenberger, &Pearscn

Ginn & company.Text: Chapter 2

A) Text: Chapter 6

A) Text: Chapter 7

A) Text: Chapter 8

d92


A) Students should demonstrate aworking knowledge of thefunctions of the angles in-cluding special angles.


A) Students should exhibit athorough competency in solvingproblems involving thefundamental identities.


A) Students should demonstrate theskills to solve the more com-plex identities involving twoangles.


A) Students should show the skillsnecessary for the solution oftriangles.


Hag)

A) Teaching Methods


V. Vectors

VI. Line Values and Graphs ofTrigonometric Functions

VII. Polar Coordinates

VIII. Approximate Numbers andLogarithms

IX. Solution of Triangles byLogarithms

A)


A)&) Refer to chap ;er 1B)

A)


A)&) Refer to chapter 1B)

A)


X. Functions A)


193 m s 51



A) Text: Chapter 1

A) Text: Chapter 3

A) Text: Chapter 4

A) Text: Chapter 12

A) Text: Chapter 14

A) Text: Anqlvtic Geometrvaugwa

IntrodtatiOlidalacaglasSchock & WarshawPrentice-Hall.

Text: Chapter 1

?94


A) Students should demonstrate theability to solve vector problemsusing trigonometric functions.


A) Students should demonstrate theability to graph the varioustrigonometric func4..sns.

3) Teacher-made test

A) Students should show the abilityto convert freely betweenrectangular and polar coordinates.

E) Teacher-made test

A) Students should manifest acomputational skill inlogarithms.


A) Students should indicate theability to solve triangles bymeans of logarithms.


A) The students should demonstratea working knoirledge of theconcept of a function.


M s 52

Concepts


XI. Coordinates and Graphs A)&) Refer to chapter 1B)

XII. The Straight Line A)


XIII. Conic Sections A)


XIV. Curve Tracing A)&) Refer to chapter 1B)

XV. Polar Coordinates

195


14 3 53



A) Text: Chapter 2

A) Text: Chapter 3

A) Text: Chapter 4

A) Text: Chwiter 5

A) Text: Chapter 6

196


A) Students should exhibit afunctional understanding of therectangular coordinate system.


A) Students should be able tosolve problems involving allaspects of the straight line.


A) Students should exhibit afunctional knowledge of theconic sections.

B) Teacher made test

A) Students should be able toplot the curves of variousalgebraic functions, in-cluding logarithmic, exponen-tialand absolute value.


A) Students should demonstratethe ability to perform con-versions, transformations, androtation of axis, with polarand rectangular coordinates.


H s

A) Teaching Methods

Concepts 3) Learning Activities

XVI. Locus Problems (Optional)

XVII. The Derivative

A)


A)

&) Refer to chapter 1

B)

XVIII. Differentiation of Algebraic A)

Functions &) Refer to chapter 1B)

XIX. Applications of the Derivative A)


XX. Integration

197

A)&) Refer to chapter 1

B)

M s 55


Resources C) People31) Places

A) Text: Chapter 7

A) Text: Chapter 8

A) Text: Chapter 9

A) Text: Chapter 10

A) Text: Chapter 11


A) Students should showfamiliarity with thebasic locus theoreMs.


A) Students should show:1) basic understanding of limits;2) the ability to find the

derivative by the deltamethod,


A) StudeAs should be able tofind the derivativcc of theprimary algebraic 'unctions.


A) Students demonstrate theability to use the derivativein the solution of practicalproblems.


A) Students should be able tofind certain indefiniteintegrals and to applydefinite irtegrals to simpleproblem solving.


198' Id s 56

"1.10111011.101.1.

Concepts

Advanced Senior Math

I. Sets and Numbers-Sets and basic notation-Subsets-Operations on sets-Integers-Rational and irrational numbers-Equality

II. The Algebra of Numbers 39 aLogical System1. The deductive system2. The real numbers3. Certain basic theorems4. Further theorems

19 9

A) Teaching MethodsB) learning Activities

A) The teaching methods in advancedsenior math are1. lecture and demonstration;2. discussion.

B) The learning activities include:1. working assigned problemsat desks:.

2. presenting assigned problemsto the class.

The first semester covers conceptswhich the students have seenprior to this course. Theseconcepts are presented, however,in greater depth. The teachermay find that he is able to movewith some rapidity and in somecases finish ahead of schedule.The difficult concepts in thesecond semester may then begiven greater time.

A)&) Refer to chapter 1

s



A) Modern College AlgebraVance- Addison-WesleyChapters l&2


A) Studmts are expected to demon-strate a grasp of the conceptsreviewed here.

B) Teacher-made testing, device

A) Text: chapters l&2 A) See above

200 M s 58

A) Teaching Methcds


III. Extensions of the Logic ofAlgebra- Addition of algebraic expressions A)

-Multiplication of algebraic &) Refer to chapter 1

expressions B)

- Divis on of algebraic expressions- Special products-Factors and factoring-Simplification of fractions-Addition of fractions-Multiplication and divisionof fractions- Integral and zero exponents-Rational exponent)-Radicals-Addition and subtraction o'if

radicals-Multiplication and division ofradicals

IV. Inequalities, Absolute Valuesand Coordinate Systems-Order axioms for the realnumbers-A one-dimensional coordinatesystem-ThP completeness property-A two-dimensional coordinatesystem-The distance formula

V. Functions and Their GraphicalRepresentation-Functions and relations-Graphical representation offunctions and relations-Graphical representation of

empirical date.

A)

&) Refer to chapter 18)

A)&) Refer to chapter 18)

201 m s 59



A) Text: chapter 3

A) Text: chapter 4

A) Text: chapter 5


A) Students should demonstratea thorough working knowledgeof the basic concepts ofalgebra.


A) Students should exhibit theability to set up a logicalsystem of coordinate algebra.


A) Students should manifest theability to discriminate betweena function and a relation.


Id s 60

A) Teaching Methods


VI. Linear and Quadratic Functions- The linear function-Arithmetic progressions-The quadratic function-Solution of the quadraticequation- Inequalities

-Relations between zeros andcoefficients of the quadraticfunction- Equations in quadratic form- Equations involving radicals-Variation-Solution of two linear equations-Algebraic solution of threelinear equations-Solution of one linear and onequadratic equation

VII. Determinants-Determinants of order two andthree- Determinants of order n

-ExpanSion,of a determinant byminors- Solution of a system of linearequations by determinants

A)&) Refer to chppter 1B)

A)


VIII. Polynomial Functions-Certain theorems A)- Graphing of polynomial function &) Refer to chapter 1- General remarks on zeros and 8)

roots-Rational roots-Irrational roots

IX. Permutations, Combinations,and the Binomial Theorem-The fundamental principle A)- Permutations &)Refer to chapter 1-Combinations B)

- The binomial theorem-The expansion of (14x)2 1A3 ids 61


Resources C) People0) Places

A) Text: chapter 6

A) Text: chapter 7

A) Text: Alapter 8

A) Text: chapter 10

"04


A) Students uhould digplay a fun,:

ti.onal understanding uhc

quadratic function.P) Teacher -mn(le test

A) Students should exhibit theability to solve determinantsof order two & three.


A) Students should demonstrate theskill to determine the rootsof various degree equations

B) Tech er -made tests

A) Students should manifest thenecessary skills required inpermutations, combinations and

probability.B) Teacher-made test

6,2

Concepts

X. Analytic Preparation- Introduc tion

-Coordinates-Directed distance-Slope of a straight line-Inequalities-Absolute value-Absolute values of sums, products,quotients, and differences-Intervals and neighborhoods-Equations of straight lines

XI. Plane Analytic Geometry-Curves and equations-Tangents and normals-Newton's method for approxi-mating roots of equations-Distance between two points-The circle-The parabola-The ellipse-.The hyperbola-Second degree curves-Invariants and the discriminant-Sections of a cone-Curve fitting using determinants

XII. Functions-Introduction-What is a function?-Equal functions. Futictiens of.,more than one variable.-Ways of combining functions-Slope of a curve-Derivative of a function-Velocity and rates


A) The teacher should procede fromthis point in the course withdctliberacion The material isquite difficult for high schoolstudents. Any true understandingof the concepts will be of greatvalue to the college boundstudent. Keep in mind that itis better to cover less materialbetter than a lot of materialpoorly.

The teaching methods are essen-tially the same:1) lecture & demonstration2) discussion.

A) At the teacher';; option, certainrefeL.oc.ea to the calculus maybe omitted during the study ofthis chapter for later presen-tation.


"0t)Ma 63




A) Elements of Calculus and Analytic A) Students are expected to demon-Geometry Thomas Addison-Wesley strate knowledge of the basicsText: chapter 1 of Analytics.


A) Text: chapter 11

A) Text: chapter 2

20B

A) Students should exhibit a compre-hensive understanding of theconcepts of Plane Geometry.


A) Students should exhibit theability to solve problemsdealing with all phases offunctions.


M s 64

A) Teaching Methods


XIII. Limits-Definition of a limit of a t.)

functicn &) Refer to chapter 1-Theorems about limits B)

-More theorems about limits-Infinity Note: The concept of a limit is

vital to a true understandingof the calculus and care shouldbe taken in the presentation.

XIV. Derivatives of AlgebraicFunctions

-Polynomial functions and theirderivatives &) Refer to chapter 1-Rational functions and their B)

derivatives-Implicit relations and deriva-tives

-The increment of a function-The chain rule for eerivatives-The differentials dx and dy.

XV. Applications-Sign of the first derivative A)Application to curve sketching. &) Refer to chapter 1-aelated rates B)

-Significance of the sign of thesecond derivative-Curve plotting-Maxima and minima th?.ory

-Maxima and minima problems-Rollo's Theorem-The Mean Value theorem

20 s 65



A) Text: chapter 3

A) Text: chapter 4

A) Text: chapter 5


A) Students should demonstrate athorough working knowledge ofthe theory of limits.


A) The students should demonstratethe ability to differentiatethe alginate functions.

B)Teacher-mde test

A) Students should exhibit theability to apply the derivative.


208 M s 66

A) Teaching Methods


XVI. Integration-Introduction-The indefinite integral-kpplications of indefiniteiltegration-3rief review of trigonometry-3ifferentiation and integrationof sines and cosines-.tree under a curve-Computation of areas as limits- Areas by calculus-The definite integral and theFundamental Theorem of integralcalculus-The trapezoidal rule for approxi-

mating an integral-Some comments on notation-Summary

A)&) Refer to chapter IB)

XVII. Applications oZ the DefiniteIntegral

-Introduction A)

-Area between two curves &) Refer to chapter 1

-Distance B)

-Volume-Approximations-Length of a plane curve-Area of a surface of revolution-Average value of a function-Moments and center of mass-Cetttroid and center of gravity

-Work

;:09Ms 67


Resources C) Peoplc..D) Places

A) Text: chapter 6

A) Text: chapter 7


A) Students should show the abilityto find both indef'nite anddefinite integra?s.


A) Students ahould demonstrate theability to apply the definiteintegral :o various pLoblems.


M s 68

Date post:	17-Jul-2020
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

ED051178.tif - ERICMaterial needed for an activity are listed with the activity description. Most...

Documents