UMBER NOllONS PRODUCED BY THE -GEORGIA EDUCATIONAL TELEVISION NETWORK TEACHER GWEN MILES STATE BOARD OF EDUCATION James S. Peters, Chairman Robert Wright, Vice-Chairman Claude Purcell, Secretary -M -E -M -B -E-R -S FIRST CONGRESSIONAL DISTRICT J. Brantley Johnson SECOND CONGRESSIONAL DISTRICT Robert Byrd Wright THIRD CONGRESSIONAL DISTRICT Thomas Nesbitt, Jr. FOURTH CONGRESSIONAL DISTRICT Donald Payton FIFTH CONGRESSIONAL DISTRICT David F. Rice SIXTH CONGRESSIONAL DISTRICT James S. Peters SEVENTH CONGRESSIONAL DISTRICT Henry Stewart EIGHTH CONGRESSIONAL DISTRICT Lonnie E. Sweat NINTH CONGRESSIONAL DISTRICT Cliff C. Kimsey, Jr. TENTH CONGRESSIONAL DISTRICT William Preston FOREWORD We are now providing more televised instruction that we hope will be of help to you in your classroom. YOU are the best authority on HOW it will help you, and in what ways you wish to use it. We are providing teacher guides like this one with suggestions that may be of service to you as you plan the best use of these lessons and fit them into the program you have planned. These guides were written by our television teachers. We think of the television teacher and the classroom teacher as being partners in the best creative teaching for the children. Television's dynamic power--Iong used in communicating other information--is now being made use of in education. It is making this a better educated world. None of us knows as much as we would like to know about it. It is a new medium and we are all learning together. We need your help and your suggestions as we seek to make the best use of our television facilities. Our aim is to make the school program more meaningful in Georgia. Our competent television teachers are well prepared to help you and the members of your class with lessons in science, mathematics, modern foreign languages, music, and Georgia history. They have time to gether up visuals that may not be readily available to you or that you may not have time to collect. This relieves you of much planning and preparation and leaves you with more time to devote to the actual teaching of the child in the classroom, and your personal teaching-andlearning contact with him. I hope you will find this teacher guide useful in your classroom work. We would be happy to have your suggestions about how our television teaching can be made more effective. If you have found some especially good ways to adapt these lessons to your pupils, let us know about it. Perhaps it would help other teachers. This is a cooperative venture; it is important that we all work together to make the best use of this new power that has come into our hands in this technological age, so that we may make learning more effective in Georgia schools. ---CLAUDE PURCELL state Superintendent of Schools GEORGIA EDUCATIONAL TELEVISION NETWORK GEORGIA STATE DEPARTMENT OF EDUCATION UPPER-MIDDLE ELEMENTARY MATHEMATICS OUTLINE TEACHER: Gwen Miles TITLES 1. Computer Mathematics 2. Other Bases 3. Astronomical Numbers 4. Operations 5. Multiplication 6. Division 7. Mathematics in Literature 8. Estimation - 9. If Then 10. Factor Trees 11. Common Factors and Multiples 12. Fractions - Who are They? 13. Operations - Rational Numbers 14. Operations - Rational Numbers 15. Decimals l6 Q Multiplication PRINCIPLE Numeration and Place Value Numeration and Place Value Place Value in the Decimal System Addition and Subtraction, Operation and Properties Operation and Properties Operation and Properties Enrichment (Application) Estimation Reasoning Number Theory: Primes, Factors, Multiples and Factorization Factorization, GCF and LCM Identifying Rational Numbers Addition on the Set of Rational Numbers Subtraction on the Set of Rational Numbers Decimal Notation for Rational Numbers Multiplication of Rational Numbers i TITLES 17. Division 18. Air Travel 19. Mathematics at the Game Ranch 20. Mathematics in Music and Art 21. cartography 22. Longitude and Latitude 23. Graphs 24. Graphs 250 Mathematicians 26. small Skiffs and Luxury Liners 27. Measurement 28. Imagination 29. Perimeter and Area 30", Space Figures 31. Constructions 320 Congruence 33. Projects in Mathematics PRINCIPLE Division of Rational Numbers Decimals, Percent Percent and Proportion Enrichment (Application) Ration and Scale Drawing Longitude and Latitude Graphs and Charts Graphs of Sets of Points andl Number Pairs Enrichment Enrichment - Measurements Measurement Non-Metric Geometry Geometry Geometry Geometry Geometry Projects for Instruction and for Fun ii GEORGIA STATE DEPARTMENT OF EDUCATION DR.. CLAUDE L. PURCELL STATE SUPERINTENDENT OF SCHOOLS NUMBER NOTIONS UPPER-MIDDLE GRADES MATHEMATICS TEACHER: Miss Gwen Miles LESSON 1-: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: COMPUTER MATHEMATICS OBJECTIVES: 1. To show ways in which a different numeration system is employed, i.e. the computer employing the binary numeration system 2. To demonstrate some kinds of problems which are programmed into the computer NATURE OF CONTENT: This lesson is to show children something about the computer. It is for enrichment and it is not intended that the binary numeration system be taught. It is to point out that it is possible to have such a system with just two numerals~ Some kinds of problems are illustrated. This lesson is not intended to launch students into a lengthy study of other bases. ADDITIONAL RESOURCE INFORMATION: 1. ComEuters - Their History, Present AEElications and Future, 1965 2. ComEuter Oriented Mathematics, National Council of Teachers of Mathematics, 1963 3. Mathematics!9 Georgia Schools 4. Reference Books 1 LESSON 2: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: OTHER BASES OBJECTIVES: 1. To investigate some of the similarities in base ten and other bases 2. To better understand base ten NATURE OF CONTENT: This lesson introduces the basic ideas in any numeration system. The principles of place value and ordered sequence are emphasized. It is suggested that classroom teachers not spend too long on other bases since this study is suppose to help students understand base ten and is not "the most important idea in "updated" mathematics. Developing operations in other bases and the extent to which a teacher carries this study is left to the discretion of the teacher. ADDITIONAL RESOURCE INFORMATION: 1. An abacus can be purchased or made to demonstrate place value 2. The How and Why wonder Book of Mathematics, 1961 3. The GIant Go'Id'en Book oTMa"t'Fi'ematics, 1960 4. NUMBER: THE LANGUAGE OF SCIENCE, Dantzig, Tobias 5. Professional Books (See Reference Chart in Principles of Mathematics ! Elementary Teachers, ETV Guide) 6. Mathematics for Georgia Schools NOTE: A bibliography for full information on suggested books and for other books on every topic can be found at the end of this manual and in the State Guide, Mathematics for Georgia Schools. 2 LESSON 3: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: ASTRONOMICAL NUMBERS OBJECTIVES: 1. To illustrate the idea of place value in base ten numeration system by use of large numbers 2. To underscore the order of numbers ("greater than" and "less than") NATURE OF CONTENT: Astronomy is used as the setting to illustrate large numbers. The ideas of place value and order are noted. This is one of many lessons in this series where basic mathematical principles are taught in an "application setting". ADDITIONAL RESOURCE INFORMATION: Note: The "follow-up" of this lesson (as in every lesson) depends on the lesson plans of the classroom teacher. Large numbers can be taught in whatever context or manner the teacher chooses. If astronomy as a area of application is used in "follow-up" the following books may be helpful. It should also be stated that the television lesson schedule does not imply that a week is to be spent on this or any other topic. 1. The Golden Book of Astronomy, 1959 2. Mathematics for Georgia Schools 3. REFERENCE BOOKS 4. What's QE There, a source book prepared by the NASA in cooperation with the U. S. Office of Education 3 LESSON 4: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: OPERATIONS AND EQUATIONS OBJECTIVES: 1. To define the operations of addition and subtraction with the set of whole numbers 2. To teach the properties of addition i.e., commutativity, associativity, and closure 3. To investigate and see if these properties hold for subtraction 4. To apply these concepts in word problems NATURE OF CONTENT: For mos~ students the properties of operations will be review. The operations of addition and subtraction are defined and the laws or properties illustrated. Word problems are employed to show how these ideas can be used to write mathematical sentences or equations and to solve them. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics for Georgia Schools 2. PROFESSIONAL BOOKS (See Reference Chart in Principles of Mathematics for Elementary Teachers, ETV Guide) 4 LESSON 5: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: MULTIPLICATION WITH WHOLE NUMBERS OBJECTIVES: 1. To define the operation of multiplication 2. To teach properties belonging to the operation of multiplication with the set of whole numbers NATURE OF CONTENT: With the set of whole numbers as our "elements" to work with, multiplication as a binary operation is defined and the properties of commutativity, associativity, closure are taught. The distributive property (called law or principle in some texts) is explored with the set of whole numbers. The distributive property relates to the operations of multiplication and addition. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics for Georgia Schools 2. PROFESSIONAL BOOKS 5 LESSON 6: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: DIVISION WITH THE WHOLE NUMBERS OBJECTIVES: 1. To define division as the inverse of multiplication 2. TO teach division as "finding the missing factor" in a multiplication problem NATURE OF CONTENT: The operation of division is explored with the set of whole numbers. Division is defined as the inverse of multiplication. with a multiplication equation there are two related division equations. The relationship between the equations is pointed out. Example of this idea: ...-/'150: 5 = 30 30 x 5 = 150 ~ 150 -;-30 = 5 It is noted that closure does not hold for the operation of division on the whole numbers. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics for Georgia Schools 2. PROFESSIONAL BOOKS 6 LESSON 7: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: MATHEMATICS IN LITERATURE OBJECTIVES: 1. To show that even in the writing of poetry mathematics is employed, and in some literature there are references to mathematics 2. To view a few examples of mathematics found in literature NATURE OF CONTENT: This is an enrichment lesson showing applications of mathematics in the writing of poetry as in such ideas as meter and feet. One interesting kind of poetry is Japanese haiku (hi'ku). Haiku is a verse-form in which Japanese poets have been working for hundreds of years. Each poem contains only seventeen syllables. The reader creates much pleasure for himself in these delightful verses. Illustrations from a famous story, Alice in Wonderland show some interesting mathematical ideas. The author, Lewis Carroll (whose real name was Charles Lutwidge Dodgson), was a teacher of mathematics. Many references to mathematics occur in Alice in Wonderland and Through the Looking Glass. Two or three of these are noted in this lesson. ADDITIONAL RESOURCE INFORMATION: This lesson requires no follow-up. There are many references to these and other selections from literature if a teacher is interested in pursuing this study. 7 LESSON 8: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: ESTIMATION OBJECTIVES: 1. To introduce concepts of estimation 2. To provide some interesting situations for word problems requiring estimation NATURE OF CONTENT: The question, "How many blocks are in this container," is asked at the beginning of the lesson. It is answered at the end of the lesson. This is just to introduce the idea of "educated" guessing and shown to lead to estimating mathematical quantities with reasonably close approximations. Situations for word problems are given which provide experiences in working with estimation and finding estimates, i.e., "We are going to sow a lawn which is 98 feet long and 47 feet wide. We know how many seeds should be sown to have grass for one square foot. How can we find quickly how many square feet of lawn we have to cover?" Other problems having to do with distance and time are illustrated, as well as problems involving the "rounding off" of large numbers. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics for Georgia Schools PROBLEMS ARE SUGGESTED FOR STUDENTS ACTIVITIES: 1. Estimate the number of gallons of milk you drink in one year. 2. Estimate the number of breaths you take in one hour. 3. Estimate the total number of days attended school by your whole class. 8 LESSON 9: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: IF - THEN OBJECTIVES: 1. To study the importance of the "if-then" relationships in mathematics 2. To show various ways of reasoning in word problem experience 3. To give some better understanding of the role of reasoning in the study of mathematics NATURE OF CONTENT: A few word problems involving deductive thinking for solutions are explored. Some of the reasoning processes which occur as one calculates using all operations are shown. Some distinction is pointed out between the words, "and", "or" and "not" as they are used in mathematics. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics for Georgia Schools 2. School Mathematics Study Group Materials 9 LESSON 10: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: FACTOR TREES OBJECTIVES: 1. To teach the following terms: a. Factors b. Multiples c. Composite Numbers d. Prime Numbers NATURE OF CONTENT: Factors and multiples are defined and illustrated. The process of factorization is stressed and one form for writing the factor of a number called "factor trees" is used. composite numbers and prime numbers are discussed. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics for Georgia Schools 2. New Texts 3. STUDIES IN MATHEMATICS VOLUME IX 10 LESSON 11: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: COMMON FACTORS AND MULTIPLES OBJECTIVES: 1. To teach the following ideas: a. Cornman Factors b. Cornman Multiples c. Cornman complete Factorization d. Greatest Cornman Factor (GCF) e. Least Cornman Multiple (LCM) NATURE OF CONTENT: Factors which are cornman to two or more numbers are shown. with set language the comparison of sets of factors and sets of multiples are pictured using the terms I union and intersection. The concepts of the greatest common factor and the least cornman multiple are introduced. The process of complete factorization is also introduced. Using this process for finding the GCF and LCM is studied. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics for Georgia Schools 2. Professional Books (See Reference Chart in guide for Principles of Mathematics for Elementary, Teachers" I Georgia ETV Programs for Teachers ,,;;;;------- - 3. Studies in Mathematics Volume IX 11 LESSON 12: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: FRACTIONS - WHO ARE THEY? OBJECTIVES: 1. To teach the meanings of a fraction 2. To clarify the terms "fraction" and "rational number" 3. To show some ways fractions are needed 4. To stress the concept of sets of equivalent fractions NATURE OF CONTENT: The approach used on the television lessons to teach the difference between "fraction" and "rational number" is much like the comparison made between "number" and "numeral". The word "rational" refers to the number or idea, while the word "fraction" is a symbol (numeral) for this kind of number. A rational number has many names or fractional representations. Furthermore, fractions (or numerals for rational numbers) fall into equivalence sets or classes. Associated with each set of equivalent fractions is exactly one rational number. A fraction is also referred to as a number pair or a pair of ordered numbers. Work with sets of equivalent fractions is stressed. Some physical models are used to illustrate this. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics for Georgia Schools 2. New Text Books 3. Studies in Mathematics Volume IX Filmstrips: 1. FRACTIONS: Gundlach, B. H. 2. FRACTION SERIES: SVE Colonial Films 12 LESSON 13: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: ADDITION OF RATIONAL NUMBERS OBJECTIVES: 1. To stress the importance of sets of equivalent fractions 2. To show addition of rational numbers using sets of equivalent fractions 3. To define the operation of addition on the set of rational numbers 4. To demonstrate why common denominators are necessary in addition of rational numbers NATURE OF CONTENT: Sets of equivalent fractions are reviewed and their importance is stressed. The operation of addition on the set of rational numbers is defined and illustrated using physical models and sets of equivalent fractions. It is emphasized that with each set of equivalent fractions there is exactly one rational number and one point on the number line. Rational numbers may be named by anyone of the fractions from the set associated with that number. The number line is used to illustrate this. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics for Georgia Schools 2. Professional Books (See Reference Chart in Principles of Mathematics for Elementary Teachers, ETV Guide) 13 LESSON 14: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: THE OPERATION OF SUBTRACTION ON THE SET OF RATIONAL NUMBERS OBJECTIVES: 1. To teach the operation of subtraction on the set of rational numbers 2. To use word problems to study addition and subtraction NATURE OF CONTENT: Subtraction as the inverse of addition is taught. These operations are represented with the number line and with physical models. Word problems are used to show a need for the operations of addition and subtraction. continued use is made of sets of equivalent fractions. Both understanding and convenience are presented as outcomes of studying this concept. Changing a fraction to an equivalent fraction by using the "property of oneil is reviewed. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics for Georgia Schools 2. Professional Books (See Reference Chart in Principles of Mathematics ~ Elementary Teachers, ETV Guide) 14 LESSON 15: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: DECIMAL FRACTIONS OBJECTIVES: 1. To introduce decimal notation 2. To teach decimals as a notation for a fraction 3. To see decimals as they are used in the world around us NATURE OF CONTENT: Decimal fractions are introduced at this point to help students see that we are still working with fractions and that decimals are a way of representing rational numbers. Place value concepts are extended to include positions to the right of the decimal, i.e., tenths, hundredths, etc. Decimals as they are used in many ways are viewed. with such ,examples as money, odometers, and the gasoline pump at the service station, decimal applications are pointed out. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics i2 Georgia Schools 2. Professional Books 3. The World Almanac, For example in the 1965 Edition, pages 486, 487 and 496 give precipitation charts in decimal notation, also wind speeds on 487. 4. Newspapers 15 LESSON 16: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: MULTIPLICATION OF RATIONAL NUMBERS OBJECTIVES: 1. To introduce the operation of multiplication on rational numbers 2. To demonstrate regions being used as models to find the product of two rational numbers represented by unit fractions 3. To use the number line to represent multiplication of rationals NATURE OF CONTENT: Physical models of regions are used to demonstrate the idea of multiplication on rationals. The number line is used to represent this also. The same pair of rational numbers will be used on these two methods as well as the method of investigating patterns using the concept of sets of equivalent fractions. Different fractions are used to represent the same pa,lr 0 f rat'lona1 n umbers l. .e.: -1 x -1, -2 x -2. After these 234 6 three methods of exploring this idea of multiplying 1 x 1 23 the same procedure will be carried out with 1 x 1 It is 3 2 stressed that multiplication is an operation on two numbers in this case rational numbers and though it is an idea or concept, this idea can be represented or illustrated with physical models. The multiplicative inverse or reciprocal of a rational number is introduced. ADDITIONAL RESOURCE INFORMATION: 1. New Texts 2. Professional Books 3. Studies in Mathematics Volume IX 16 LESSON 17: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: THE OPERATION OF DIVISION ON THE SET OF RATIONAL NUMBERS OBJECTIVES: 1. To teach the concept of a reciprocal 2. To introduce the operation of division on rational numbers NATURE OF CONTENT: The concept of every rational number having a reciprocal is repeated. It is stressed that the rational numbers have this important property not possessed by either the set of counting numbers or the set of whole numbers. The reciprocal (sometimes called the multiplicative inverse) of a,number is the numbe~ by wichsit Tust be ~ultkpliI~ to glve a product of lr l.e., - x - = - = 11 - x - = --= I; 511 4 3 12 - 3 x 1 = 3- = 1. 33 The reciprocal of a is b r and the reciprocal b a of b is S ab By employing this idea of a reciprocal and the property of one r that is the number 1 is the multiplicative identity (any number multiplied by one gives a product of that same number)r the operation of division on rational numbers is explored. Division for the set of whole numbers is defined as the inverse of multiplication. If mathematics is to be consistent division on the set of rational numbers must be the inverse of multiplication. The following is an example of carrying on this discovery: ~7-~=( ) ( )x ~ - ~ (! x ~\ x ! = 1 ~ 1) 4 2 Division means "what number multiplied by ! equals 1 " 4 2 17 LESSON 17: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLEs THE OPERATION OF DIVISION ON THE SET OF RATIONAL NUMBERS CONTINUED: b x4 x1\= 1 2 \ 1 4) 2 bx 1 =1 2 2 By the associative property ~ ::- By the reciprocal property and the property of one this (~ x ~ is the number which multiplied - by 41 = 21' It is 'more precise to say "multiply by the reciprocal of the divisor" than to say "invert the divisor and multiply". This is one way we justify the rule for the division on rational numbers. This division problem is represented by regions and blocks. The question asked is, "how many fourths are in one-half?" This is further illustrated with the number line. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics!EE. Georgia Schools 2. New Texts 3. Professional Books 17 a LESSON 18: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: DECIMALS, PERCENT AND AIR TRAVEL OBJECTIVES: 1. To stress the fact that decimal fractions are numbers and therefore ideas, but these ideas can be applied in many situations 2. To use air travel as a format for such a situation NATURE OF CONTENT: That numbers are ideas and abstract makes them useful. Decimal notation or decimal names for rational numbers have many applications. Air travel has been selected as a topic for word problems involving decimals. A special kind of rational number, onc whose denominator is 100, called percent is also taught in this application context. -For example, the air line distance between Atlanta, Georgia and Denver, Colorado is 1,212 miles. If a plane has flown 606 miles what percent of the trip has it traveled so far? The proportion method is used to solve this problem. 606 1212 = N 100 l2l2N = 60600 N = 50 and -5= 0 100 50"~ ADDITIONAL RESOURCE INFORMATION: 1. Mathematics i Georgia Schools 2. New Texts 3. School Mathematics Study Group Material 4. The World Almanac 18 LESSON 19: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: MATHEMATICS AT THE GAME RANCH OBJECTIVES: 1 0 To reinforce work with percent and proportion 2. To present word problems as applications of percent and proportion with the setting at the Game Ranch at Stone Mountain, Georgia NATURE OF CONTENT: Applications of uses for percent and proportion are given through word problems. The setting for this lesson is the Game Ranch at Stone Mountain, Georgia. This place was chosen as an interesting situation where such problems as the percent of certain foods in the diet of the rabbits or deer can be explored. Other kinds of word problems are illustrated and solved o ADDITIONAL RESOURCE INFORMATION: 1 0 Mathematics for Georgia Schools 2. New Texts and Old Texts 3. Newspapers and Magazines are an excellent source for finding percent notation 4. School Mathematics Study Group Materials 19 LESSON 20: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: MATHEMATICS IN MUSIC AND ART OBJECTIVES: 1. To show other applications of ideas of mathematics 20 To point out that both the musician and the artist employ mathematical ideas NATURE OF CONTENT: One of the many ideas of mathematics which are needed by the musician is demonstrated, i.e., fractions as they are related to tones or vibrations of a string's length. Frequency, tones, and keys are discussed, and some of'the simple mathematical relationships are brought out. The artist uses two mathematical ideas which are illustrated briefly, namely, perspective and scale drawing. ADDITIONAL RESOURCE INFORMATION: The extent of IIfollow-up ll for this lesson will depend on each teacher's interest and curriculum. It is one of the lessons in this series which may be called II special interest II and no follow-up is required. 20 LESSON 21: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: CARTOGRAPHY OBJECTIVES: 1. To show that the map-maker employs the mathematical ideas, ratio and scale drawing NATURE OF CONTENTl with a brief definition of the term "ratio", this lesson is primarily for enrichment o Some of the problems and procedures of the map-maker are investigated. It is demonstrated that the cartographer needs the ideas of ratio and scale drawing. The application of these ideas is underscored. At this writing it is anticipated that this will be done "on location", where aerial photographs are used to make maps. ADDITIONAL RESOURCE INFORMATION: 1. Atlas 20 Geography Books 3. Globe 4. Reference Books 5. Road Maps 6. ~ Giant Golden Book f Mathematics, Irving Adler 7. weather Bureau 21 LESSON 22: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: LONGITUDE AND LATITUDE OBJECTIVES: 1. To teach the terms, longitude and latitude 2. To show the use to which navigators put these ideas NATURE OF CONTENT: The terms longitude and latitude are explored. Some of the problems of the navigator are investigated. This topic leads easily into a study of graphs and charts. Having looked at some of the problems of the cartographer, some practice is suggested for reading maps and locating points on maps and globes. ADDITIONAL RESOURCE INFORMATION: 1. Atlas 2. Maps 3. Reference Books 4. The Giant Golden Book.f Mathematics, Irving Adler 5. The World Almanac (for such information as longitude and latitude of cities, the altitudes are given also) 22 LESSON 23: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: GRAPHS OBJECTIVES: 1. To note ways in which graphs are used (applications in interesting situations) 2. To illustrate various kinds of graphs NATURE OF CONTENT: Different kinds of graphs and their uses are illustrated. Some graphs studied will be pictographs l bar graphs l line graphs and circle graphs. The applications of graphs are investigatedo These ideas lead to the development of graphs of sets of points on the number line and graphing pairs of numbers. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics ! Georgia Schools 2. New Texts 3. School Mathematics study Group Materials 40 ~ Giant Golden Book f Mathematics l Irving Adler 5. Mathematics for Everyone I Howard F o Fehr and Max Ao Sobel 23 LESSON 24: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: GRAPHS OBJECTIVES: 1. To introduce graphing sets of points on a number line 2. To introduce graphing pairs of numbers NATURE OF CONTENT: Graphing sets of points on the number line is introduced. It is stressed that the graph of a number is a point on the number line. When a point is "marked" on the number line to represent a number, that number has been graphed", For example, graphing the number 2: o ~ 3 The following is a graph of all natural numbers greater < < than one and less than five: The notation for this graph is: Where N is a Natural number, 1 N S fl. ., , f o 1 Z 3 4- S- Graphing pairs of numbers is introduced. Ordered pairs of whole numbers are used to show that one point repre- sents a number pair and a number pair is associated with only one point. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics for Georgia Schools 2. New Texts 3. School Mathematics Study Group Material 4. The Giant Golden Book of Mathematics, Irving Adler 5. Mathematics!..;:. Everyone, Howard F", Fehr and Max Ao Sobel 24 LESSON 25: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: MATHEMATICIANS OBJECTIVE: 1. To stimulate an interest in reading biographies of mathematicians NATURE OF CONTENT: At this writing it has not been determined how many biographies or brief sketches of the lives of mathe- maticians will be attempted. Possibly some interesting events in the lives of only two people will be illus- trated. There are many exciting biographies for stu- dents to read if they are interested. Any assignment in this study will be left to the classroom teacher. This lesson could be used as a special interest lesson and not spend muchitime in follow-up. Several inter- esting people are considered for such a study i.e., John Napier (1550 - 1617), Omar Khayyam ( ? - 1123), Lise Meitner (1878 - ), Enrico Fermi (1901 - 1954), Archimedes (287 B.C o - 212 B.C.), Isaac Newton (1642 - 1727), Carl Friedrich Gauss (1777 - 1855), Albert Einstein, (1879 - 1955) oeo ADDITIONAL RESOURCE INFORMATION: 1. Makers 2t. Mathematics, Hooper, Alfred 2. ~ 2t. Mathematics# Bell, Eo To 30 Number, ~ Language 2t. Science, Dantzig, Tobias 4. .Qf ~ ~ Numbers, Muir, Jane 5. Reference Books 6 0 The Great Mathematicians, Turnball, Herbert Wo 25 LESSON ~6: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: SMALL SKIFFS AND LUXURY LINERS OBJECTIVES: 1. To emphasize the fact that with small boats or ocean liners I the mathematical idea of measurement is important 2. To use this application to point out needs for the following kinds of measurements: a. Weight d. Time b. Volume e. Area c. Linear f. Distance NATURE OF CONTENT: with small boats and large ships it is shown that the idea of measurement is important. Another enrichment lessonl this program emphasizes that many kinds of measurements are employed in the building and sailing of watercraft. It is intended that this lesson will be a stimulus for students to think of other ways measurement is applied to the operation of shipsI and to think of other topics where measurement is necessary" ADDITIONAL RESOURCE INFORMATION: 1. Books on Sailing 2" It t g About Timel Miriam Schlein 3. Mathematics!2E.. Georgia Schools 4. Reference Books 5. Timekeeping Through the Ages l Uo So Department of Commerce I National Bureau of Standards l Washington I D. Co 6" Wonderful World .f Mathematics l Lancelot Hogben 26 LESSON 27: ETV - NUMBER NOTIONSjUPPER-MIDDLE MATHEMATICS TITLE: MEASUREMENT OBJECTIVES: 1. To teach the concepts of measurements and approximation 2. To define the terms "precisionlt and "possible error" in the measuring process 3. To present the necessity for standard units NATURE OF CONTENT: It is brought out that we measure properties of "things", never the things themselves. Properties of things refer to observable characteristics i.e., length, volume, area, weight~ Numbers assigned to objects in the measuring process are called measurements. There is always an error in measurement. Any measurement is approximate no matter what instrument is used. The term precision does not refer to care taken while making a measurement but to the size of the divisions used on the measuring instrument. The smaller the size of the unit or division of unit the greater the precision. The process of "rounding off lt is discussed. The need for standard units is stressed. ADDITIONAL RESOURCE INFORMATION: 1. Fundamental Concepts .2!. Elementary Mathematics, Brumfiel, Eidholz, and Shanks 2. Mathematics!2!: Georgia Schools 3. Studies in Mathematics Volume IX 4. Other pr~essional Books (See R;ference Chart in Teacher Aid for Itprinciples of Mathematics for Elementary Teachers", ETV Course for teachers) 27 LESSON 28: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: IMAGINATION (NON-METRIC GEOMETRY) OBJECTIVES: 1. To introduce some basic ideas and terminology in non-metric geometry: undefined terms1 (point, line, plane) ,space, curve, segment, ray, closed curve, simple closed curve, special closed curves NATURE OF CONTENT: The student is made aware of the geometric concepts of point, line and space. It is emphasized that marks on chalkboards and paper or points of a pencil merely represent the idea "point". Representations of basic properties of points, lines and planes are illustrated. Other terms brought out are: curves, closed curves, simple closed curves and some special closed curves to which names have been given i.e., triangle, square, rectangle, quadrilateral, polygon, and circle. ADDITIONAL RESOURCE INFORMATION: 1. Fundamental Concepts of Elementary Mathematics, Brumfiel, Eicholz, and Shanks 2. Mathematics !2 EVeryone, Howard F. Fehr and Max Ao Sobel 3. Mathematics ~ Georgia Schools 4. Professional Books 5. School Mathematics Study Group Material - - 6. Studies in Mathematics Volume IX 70 The Giant Golden ~ of Mathematics, Irving Adler 8. The Story ! Mathematics, By and Englehart Ruchlis 28 LESSON 29: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: PERIMETER AND AREA OBJECTIVES: 1. To teach the basic thinking in "finding" the perimeters of rectangles, squares, triangles 2. To explain the meaning of the circumference of a circle 3. To introduce the process for finding the measure of the interior region of a simple closed curve NATURE OF CONTENT: Simple closed curves are reviewed. The total length or measure of the curve is called the perimeter. Appropriate units for measuring are discussed. Finding perimeters of squares, triangles and rectangles is demonstrated o The meaning of the circumference of a circle is explored but the formula is not taught at this time. The basic principles of finding the measure of the interior region of a simple closed curve are taught. The name we give this measure of a plane region is the area. Unit squares are used to demonstrate the ideas involved in finding areas. The measure of a unit square is one square unit. It is pointed out that we do not multiply units by units for example, feet times feet to obtain square feet. Our formulas are the result of generaliZing the basic idea of using unit squares. Multiplying the numbers which are the measure of the length and width of the rectangle is a short-cut way of finding the number of square units. It is also remembered that all measurement is approximate. Actually in the formula = for finding the area of a rectangle A w X 1, w represents the number of unit strips and 1 represents the area of each of these strips 0 In addition to the unit squares and unit strips, the grid is used to illustrate the measure of the region of a simple closed curve o 29 LESSON 29: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: PERIMETER AND AREA CONTINUED: ADDITIONAL RESOURCE INFORMATION: 1. Fundamental Concepts .f Elementary Mathematics, Brumfiel, Eicholz and Shanks 2. Mathematics for EVeryone, Howard F. Fehr and Max Ao Sobel 3. Mathematics for Georgia Schools 4. Professional Books 5. School Mathematics Study Group Material 6. Studies1 Mathematics Volume IX 7. The Giant Golden Book f Mathematics, Irving Adler 8. ~ stOry .2f Mathematics, Hy and Englehart Ruchlis 29 a LESSON 30: ETV - NUMBER NOTIONSjUPPER-MXDDLE MATHEMATICS TITLE: SPACE FIGURES OBJECTIVES: 1. To teach the basic definition of three-dimensional or space figures 2. To view representations of these geometric shapes and to view objects in the world around which remind us of these shapes NATURE OF CONTENT: Three dimensional figures (sometimes called solids or space figures) are figures whose set of points do not all lie in one plane. We have said that a line is a set of points, a plane is a set of points, a subset of space, now it is pointed out that space figures are a set of points all of which are not in the same plane. Clarification of the use of the word "solid" to mean space figures is achieved with demonstrations of cubes and rectangular prisms. Objects which can be observed remind ~ of these geometric ideas. This is to help students understand the relationship between the physical world and geometry. The shapes observed relate to: cubes, rectangular prisms, cylinders, cones, spheres, pyramids and triangular prisms. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics 1.;:. Everyone, Howard F <> Fehr and Max Ae> Sobel 2. Mathematics!9 Georgia Schools 3 New Text Books 4. School Mathematics Study Group Material 5. The Giant Golden Book .f Mathematics, Irving Adler 6. The How and Why Wonder Book .f Mathematics, Esther He>, and Harold J. Highland 7 ~ story .f Mathematics, Hy and Englehart Ruchlis 30 LESSON 31: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: CONSTRUCTIONS OBJECTIVES: 1. To introduce the concept of constructions in geometry 2. To introduce copying segments, angles, triangles NATURE OF CONTENT: Drawing rays, lines, angles and circles with the use of a compass is demonstrated. The steps in these basic simple constructions are surveyed. The classroom teacher will determine (as always) how much time students will spend on construction. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics for Georgia Schools 2. New Texts 31 LESSON 32: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: CONGRUENCE OBJECTIVES: 1. To introduce the concept of congruence (pronounced con'gruence these days) 2. To demonstrate using a compass to compare two line segments and two angles 3. To look at some congruent triangles and discuss congruence of triangles NATURE OF CONTENT: Congruence will be defined and demonstrated .Mith line segments, angles and triangles. The compass is used to compare two angles and two line segments. The extent of studying the congruence of triangles is brief because of time but remember that the purpose of the television lessons is just to introduce or enrich ideas, the teaching occurs in the classroom. ADDITIONAL RESOURCE INFORMATION: 1. Mathematics!3 Georgia Schools 2. New Texts 32 LESSON 33: ETV - NUMBER NOTIONS/UPPER-MIDDLE MATHEMATICS TITLE: PROJECTS IN MATHEMATICS OBJECTIVE: 1. To show some of the kinds of projects which students can make for instruction and some just for fun NATURE OF CONTENT: Projects which students can make are viewed. Such projects include: space figures made of plaster-of-paris and lumber scraps, abaci, place value charts, geometric designs made with straight edge and compass, games and puzzles. At this writing the plans are to have guests students who have made some of these objects. ADDITIONAL RESOURCE INFORMATION: 1 ~ Magic House ~ Numbers I Irving Adler 2. The Giant Golden Book ~Mathematics, Irving Adler 3 0 The ~ ~ Why Wonder ~ .2i Mathematics, Esther Ho I and Harold J.. Highland 33 SUGGESTED BOOKS FOR STUDY AND INTEREST NOTE: Professional books recommended very highly are indicated with an asterisk. This is not intended to be a complete list of professional or of other books. Adler, Irving~. The Giant Golden Book of Mathematics, Rockefeller Center, No Yo 20, N.Y o , Golden Press, 1960. Adler, Irving. MaSic House ~ Numbers, New York, 501 Madison Ave., Signet Key Books, 1957. Adler, Irving. The ~ Mathematics, New York, Mentor Book, New American Library of World Literature, 1960. Banks, J o Ho~ Learning and Teaching Arithmetic, N. Y., Allyn and Bacon, 1964. Bell, Eo T 01 ~ ! Mathematics, New York, Simon and Schusterm, 1937. *Brumfiel, Eicholz and Shanks. Fundamental ConceEts of Elementary Mathematics, Reading, Massachusetts, Addison-Wesley Publishing Co., Inc., 1962. Crouch, Ralph and Baldwin, George. Mathematics for Elementary Teachers, New York, John Wiley and Sons, 1964. Dantzig, Tobias. Number, ~ Languase ~ Science, Garden City, No Y., Doubleday Anchor Books, Doubleday and Co., Inc., 1954 0 Drooyan, Irving and Hadel, walter. ~ Prosrammed Introduction to Number Systems, New York, John Wiley and Sons, 1964. Evans, Trevor. Fundamentals f Mathematics, Englewood Cliffs, No J o , Prentice-Hall, Inc., 1959. Fehr, Howard F 0' and Sobel, Max A., Mathematics ~ EVeryone, New York, Ridge Press Pocket Books, Inc., 1963. Hacker, Barnes and Long. Fundamental ConceEts f Arithmetic, Englewood Cliffs, N.Jo, Prentice-Hall, Inc., 1963. *Heddons, James Wo, Today's Mathematics: ~ Guide ! conceEts and Methods in Elementary School Mathematics, Science Reasearch Associates, Inc., 19640 Highland, Esther Ho and Harold J., The ~ and Why Wonder Book of Mathematics, New York, Wonder Books, Inc., 1961. I SUGGESTED BOOKS FOR STUDY AND INTEREST CONTINUED Hogben, Lancelot. The Wonderful World of Mathematics, Garden, city, No Y., Garden City Books, 1955. Hogben, Lancelot. Math in ~ Making, New York, Doubleday and Co. Hogben, Lancelot. Mathematics!E. ~ Millions, New York, W. We> Norton Co., 1955. Hooper, Alfred. Makers f Mathematics, New York, Modern Library Paperback, Random House, 1948. Howard, Charles F. and Dumas E., Basic Procedures ~n Teaching Arithmetic r Boston, Do C. Heath and Co., 1963. James, Glenn and Robert C. James (eds). Mathematics Dictionary, New York, D~ Van Nostrand Co., Inc., 1959. Keedy, Marvin. A Modern Introduction to Basic Mathematics, Reading Massachusetts, Addison-Wesley Publishing Co., Inc., 1963. Kr amer, Edna. The Main stream of Mathematics, New York, A Premier Book, Fawcett World Library, 1961. Kenyon, Faymond G., ~ ~ Learn About Calculators and Computers, Harper and Brothers, 1961. Lauber, Patricia. The stOry of Numbers, New York, Random House, 1961. *Merserve, Bruce E~ and Sobel, Max: A., Mathematics i2E Secondary School Teachers, Englewood Cliffs, No J., Prentice-Hall, Inc., 1962. Muir, Jane. Of ~ and Numbers (story 21. Great Mathematicians), Dodd Mead and Company, 1961. Ohmer, Merlin M., Aucoin, Clayton V., and Cortez, Marion J., Elementary contemporary Mathematics. Blaisdell Publishing Co., 1964. Osborn, DeVault, Boyd, Huston. Extending Mathematics Understanding, Columbus, Ohio, Charles E. Merrill Co., 1961. *Peterson, John A. and Hashisaki, Joseph. Theory of Arithmetic. New York, John wiley and Sons, Inc., 1963. II SUGGESTED BOOKS FOR STUDY AND INTEREST CONTINUED Ruchlis, Hy and Englehart. The story of Mathematics. New York, Harvey House Publishers, 1958. Shipp, Donald Eo and Adams, Sam. Developing Arithmetic .Concepts and Skills, Englewood Cliffs, No J., Prentice-Hall, Inc., 1964. *Studies in Mathematics Series Volume IX. (SMSG) A Brief Course ~ Mathematics !3 Elementary School Teachers, 367 Pasadena Ave., Pasadena, California, Ao C. Vroman Co., 1963. f Schaaf, Wm. L~, Basic Concepts Elementary Mathematics, New York, John Wiley and Sons, Inco, 1960 0 Spitzer, Herbert F., The Teaching ! Arithmetic, Boston, Houghton Mifflin Co., 1961. Spreckelmeyer and Mustain. The Natural Numbers: Thinking with Numbers, Boston, Do Co Heath and Co., 1963. Swenson, Eo J.~ Teaching Arithmetic ! Children, MacMillan Publishing Co., 1964. Swain, Robert L., Understanding Arithmetic, New York, Holt, Rinehart and Winston, 1960. - - ,;;,"'----------- Turnbull, Herbert W., The Great Mathematicians, New York, Simon and Schuster, Inc., 1962. .,"'--- Ward, Morgan and Hardgrove, Co E., Modern Elementary Mathematics, Reading, Massachusetts, Addison-Wesley, 1964. Wyler, Rose and Ames, Gerald. The Golden Book of Astronomy, New York, Golden Press, 1959. Young, Frederick, H., Digital Computers and Related Mathematics, Boston, Massachusetts, Ginn and Co., 1961. Youse, Bevan K., Arithmetic: ~ Modern Approach, Englewood Cliffs, No J., Prentice-Hall, 19630 The World Almanac. New York World Telegram and The Sun, 125 Barclay st., New York IS, No Yo, 1965. III SUGGESTED BOOKS FOR STUDY AND INTEREST CONTINUED YEARBOOKS : Mathematics Enrichment for the Grades, Mathematics Enrichment for the High School. Published by the National Council of Teachers of Mathematics, 27th and 28th Yearbooks, 1963u Topics in Mathematics for Elementary School Teachers, 29th Yearbook, 1964. Published by the National Council of Teachers of Mathematics, 1201 Sixteenth st., N~ W~, Washington 6, D. C~ PERIODICALS: *The Arithmetic Teacher, National Council of Teachers of Mathematics, 1201 Sixteenth st., N~ W., Washington 6, D~ CQ The Mathematics Teacher, National Council of Teachers of Mathematics, 1201 Sixteenth st., N. W., Washington 6, D~ C~ BOOKLETS AND PAMPHLETS FOR PUPIL'S READING: THE AMAZING STORY OF MEASUREMENT, Lufkin Rule Co., 1730 Hess Ave., Saginaw, Michigan (free) FROM ABACUS TO MONROE, Monroe Calculating Machine Co., Oakland 8, California, (free) FROM OG TO GOOGOL, MarChant Calculatin 1achine Co., Oakland 8, California, (free) HOW LONG IS A ROD, Ford Motor Co., Dearborn, Michigan, (free) NUMBER STORIES OF LONG AGO, Smith, David Eugene, National Council of Teachers of Mathematics, 1201 Sixteenth sto, N~WQ' Washington 6, D. C. (75 cents) NUMBERS AND NUMERALS, Bureau of Publications Teachers College, Columbia University, New York NUMBERS AND NUMERALS, Illinois State Department of Educatio~4 Springfield, Illinois SETS g SENTENCES, AND OPERATIONS, Johnson D., and Glenn, Wm .. Ii. Webster Publishing Co., Atlanta, Georgia, 1960 IV SUGGESTED BOOKS FOR STUDY AND INTEREST CONTINUED STORY OF FIGURES 1 THE I Burrough Adding Machine Co., 6071 Second Blvd., Detroit 32, Michigan TIMEKEEPING THROUGH THE AGES, United States Department of Commerce, National Bureau of Standards, Washington, Do Co (free) UNDERSTANDING NUMBERS: THEIR HISTORY AND THEIR USE, Jones, Po So, Ulrich1s Bookstore, 547 East University Ave., Ann Arbor, Michigan, (deals with film set of same title) YOU AND TIME (The Fascinating story of Timekeeping) Bulova Watch Co., Inc., Bulova Park, Flushing 70, New York v