Mathematics and Society - OS X Lion Server

MATHEMATICS INSCIENCE AND SOCIETYPlacement Guide for Tabbed DividersTab LabelFollows page:DIDACTiCS 12Teaching for Transfer 12Teaching via Problem Solving 20Teaching via Lab Approaches 42Middle School Students 56CLASSROOM MATERIALS 70Mathematics and ASTRONOMY 70ASTRONOMY GLOSSARY 120Mathematics and BIOLOGY 126BIOLOGY GLOSSARY '" 172Mathematics and ENViRONMENT 174Popu lation 188Food 204Air 210Water 222Land and Resources 230Energy 240ENVIRONMENT GLOSSARY 248Mathematics and MUSIC 254MUSIC GLOSSARY 322Mathematics and PHYSICS 326PHYSICS GLOSSARY 368Mathematics and SPORTS 370SPORTS GLOSSARY 432ANNOTATED BIBLIOGRAPHy 434SELECTED ANSWERS 460

(I : 25 R tJl. "SHOP CLASS)-~-In a recent nation-wiae testing, fewer thanthat two eighth-notes equal a quarter-note.30% of the 13 year olas knew(NAEP Newsletter, 1974)"Transfer" refers to the influence of learning upon later performance or learning.A good case can be made for the position that one of our most important aims ineducation is transfer, since what goes on in the classroom may be worthless if ithas no effect in subsequent" classroom work, outside the classroom or in later life.Yet, from a conversation with an industrial arts or home economics teacher, it mightseem that students must regard circles and fractions encountered outside of themathematics class as novel and mysterious ideas. This section reviews the principlesof teachin~ for transfer (cf. Ellis, 1965).GENERAL GUIDELINES IN TEACHING FOR TRANSFERLearning and practice should resemble the situations to be encountered later asmuch as possible;For example, Biehler [1971] points to the importance of dressrehearsals or scrimmages in team sports in preparing for the actual events.If weexpect our students to be able to find volumes of rectangular solids, then we likelyshould do more than give them length, width and height measurements and a formula.Weshould also have them identify rectangularsolids for which they must measurethe lengths, widths and heights,perhaps even using carpenters' rulersor "real" tape measures.Dealing with,say, a real faucet (see Fix That Leakin the MATHEMATICS ANDTHE ENVIRONMENTunit is likely to have more carry-over13

DIDACTICSTEACHING FOR TRANSFERthan always working with supplied data.If we want our word problems to be morerealistic, we should have some which require sifting through extraneous informationor which involve insufficient data.learning (by doing) than reading about making a batch.Preparing a batch of cookies may lead to moreIt may be worthwhile to baseexercises on newspaper advertisements, mail order catalogues, record club brochures,bank statements or credit card bills.Work for sufficient learning of the material which is to be transferred.what does "sufficient" mean?Hence,That's the toughie.It is difficult, of course, in a given learning situation to specify preciselyhow much practice is desirable on a specific task; nevertheless, agood rule of thumb would be to have students receive as much practice asis feaqible considering the restraints imposed by the various activitiesin the classroom. l"n additi'on, the teacher 1-8 somewhat free to be seleet,liein the degree of emphasis placed on various topics. Perhaps greateremphasis could be placed on those topics that are known to be necessaryfor the mastery of subsequent course work. [Ellis, 1965, p. 71]it seems to come down to identifyingthose skills, processes, conceptsand principles which are the mosttransferable to the later study of mathematicsand its applications, and thenbeing certain to give them "sufficient"attention in class.For example, wemight deemphasize volume formulas infavor of some determination of volumesby direct filling of containers withoHMM, W>iICf-I Of"\\{E'OE: 'Ol-\OULD' IEMPI-IA'SIZE?But>\\STORY OF METR\C SYSTEMWORK WITH ARBITRARY Ut-.lITSAPPLlCA,IOWSCOWVER'S,\OIJ'2>FORMULAScubes or by pouring water or corn meal or puffed rice from standard containers.types of activities involve a measuring of volume and also put into practice the generalprinciple of measuring by direct comparison with a unit.TheseOr, we might present discoverylessons for some topics in order to focus on problem-solving processes as well asthe particular topic.Shulman summarizes the research by noting, "Discovery-learningapproaches appear to be superior when the criterion of transfer of principles to newsituations is employed." [1970, p. 58]Mathematics Resource Project's Geometry and Visualization.See GoaZs Through Discovery Lessons in theUse a variety of examples when teaching concepts and principles.The Teachingof Concepts in the Mathematics Resource Project's Geometry and Visualization givessome of the reasons and techniques for using a variety of examples(and non-examples).The understanding of a concept or principle can be strengthened, enriched and ((\(14

DIDACTICSTEACHING FOR TRANSFERclarified by a diversity of examples and settings, and such a variety alerts thestudent to the fact that the concept or principle can arise in many different contexts.Such a variety of examples not onlygives them additional experience withthe concept but also enables then, tosee many situations where the idea canbe used.For example, over a period of years the work with fractions should includeexposure to the several models for fractions (see the commentary to FRACTIONS:Concepts, in the resource Number Sense and Arithmetic Skills).The learner's backgroundmust be kept in mind, of course;introducing all possible settings on astudent's first encounter with a newconcept or principle would be illadvised.For the concept of area, studentsshould work with areas of planeregions having curved boundaries andwith surface areas of space figures, aswell as the usual polygonal regions.Gagne advises:BUT THAT DOC~10THAVE: AREA - Ot-) LYSTRAIGI-H T\.\II\lG":> DO.The more varied these (situations) oan be made, the more useful will thelearned oapability beoome. At lower eduoational levels, this variety maybe aohieved by deliberate use of a whole range of natural objeots and evenevents in the olassroom or on field trips. At higher (oollegiate) levels,the funotion of providing oontextual variety oan be largely performed byverbal oommunioation, of the sort that may take plaoe in a "disoussiongroup," for example. [1970, p. 339]Bring out the important features in a learning situation.Point out, for example,that the class' work with arbitrary units is to show that the unit of measure~ment can be arbitrary--not because anyone really cares how many paper clips wide adesk-top is.Although language is occasionally over-used and its role in learning isnot well understood, it can be utilized in verbalizing the learned rules and thespecial features that are likely to transfer.to be an aid:Just using the word for an idea seems"Labeling helps us to distinguish important features of a task,although we are not entirely sure whether this is due merely to increased attentiongiven to these features or whether it is due to the label itself." [Ellis, 1965,p. 72JDo not expect much transfer unless the students understand the general principles.This caution is much like the references to "sufficient" learning above.,15

DIDACTICSTEACHING FOR TRANSFEREmphasis on underlying key ideasshould be productive for transfer.The principle that a plane or a spaceregion can be measured by "cutting"it into pieces and then finding thesum of the measures of the piecescan be used in different waysthe figure to the right).(as inBruner describesthe importance of general principlesin this excerpt:. . (the handling) of a subject should be determined by the most fundamentalunderstanding that can be achieved of the underlying principlesthat give structure to that subject. Teaching specific topics or skillswithout making clear their context in the broader fundamental structureof a field of kn07iJledge is uneconomical in several deep senses. rn thefirst place, such teaching makes it exceedingly d~~fieult for the studentto generalize from what he has learned to what he will encounterlater. rn the second place, learning that has fallen short of a graspot genePat principles has little Y'e7iJard ~'n terms of intellectual exoitement.The best way to oreate interest in a subjeot is to render it worthknowing, which means to make the kn07iJledge gained usable in one's thinkingbeyond the situation in whioh the learning hasocCJUl>:i>ed . . • [1960,p. 31, emphasis added]Emphasize applications."Teachers set the stage for tPansfer paPtly by pointingout that the students should expeot their new learning to transfer." [Cronbach,1963, p. 322] Measurement topics are perhaps among the easiest to find applicationsfor; notice how many of the Mathematics in Science and Society activitiec involvemeasurements of some sort.Besides using the wealth of material in these units,looking at textbooks in other subject fields and talking to teachers in these fieldscan give ideas for other applications.Students can be encouraged to bring inexamples of applications of mathematics (and should be reinforced for doing so).Bulletin boards, special proj ects, guest speakers, field trips--all can focus onapplications.Since students tend to attach importance to what they are tested on,it might be productive to include application items on quizzes.Applications within mathematics are also important; not only do they reviewand reinforce other mathematics, but they also show the interrelatedness of mathematicalideas.Commutativity is an idea that can appear in geometry (combinations(((16

DIDACTICSTEACHING FOR TRANSFERof some motions) as well as with somearithmetic operations. There often isa geometric way to represent a numericaloralgebraic topic, and vice versa.Many problems allow different sorts ofsolutions ("Did anyone solve this oneby proportions?" or "Did anyone use agraph?"). Again, call these solutionsto the students' attention to maximizetransfer of the mathematical ideas andproblem-solving techniques.Be careful in situations where negative transfer might occur. Negative transfer,in which one learning interferes with another learning, is most common whensimilar situations call for different responses. For example, the length and thewidth of a rectangle are used in finding both the perimeter of the rectangle andthe area of its region. Hence, one could predict that students might get perimeterand area calculations mixed up. Some of the difficulty with percent word problemslikely stems from the similarity of the word descriptions. ."Of' means multiply"may be taken too literally. Hidden impressions like "dividing always gives a smallernumber" or "squaring always gives a larger number" may be misleading when dividingor squaring fractions. Similar vocabulary words with different meanings (e.g.,polygon and polyhedron) are often misused. Paying special attention to these similaritiesby pointing them out and contrasting the appropriate responses may give abetter performance.SUMMARYTransfer does not seem to take place automatically.shouldResearch suggests that we• use settings as "real" as we can make them,• identify and emphasize the key concepts and principles,• work in as many different examples of applications as we can,• lay stress on the notion that students should expect mathematics to pop up inlots of places, and• be alert to places where negative transfer may occur.17

DIDACTICSTEACHING FOR TRANSFER1. What are some applications (within or outside mathematics) of each of thefollowing?a. commutativityb. the problem-solving process, draw a diagramc. the volume of a prismd. exponents(2. "No matter where you turn in modern life, you will find mathematical problems.Often the best assignment is to have the student • • . find • • • applicationsin his environment. Then he can decide what data to collect and what relationshipsto explore." [Johnson and Rising, 1972, p. 279] Such assignments wouldseem to foster a transfer attitude. Give five examples that you feel your studentscould come up with.3. What are some examples of questions which might induce an attitude of "how canthis be used?" or "where else in math have we used this idea?"?4. (Discussion) Since time is so restricted, we must choose which topics to coverand which to omit. Suppose you have time for only one of each in the followingpairs. Which one would you choose (on the basis of transfer--not whether "they'll~~t i;:t;~:;:~~:;y~;i~;;~:~:~;~~;::~ro~~~:~~~t~~n: ~~g~~~el~~~l~h;~: ~~gpmultiplication factsc. more work on whole number division, or work with prime numbersd. percents over 100%, or percents less than 1%e. perimeter and area, or more work with fraction addition and multiplication(5.6.(Discussion) Two views of transfer are described below. Psychologists, eitherbecause of research or because of impracticality, have rejected them as profitableapproaches. [Cf. Biehler, 1971, pp. 265-267.] What are your feelings aboutthem?a. The way to increase the powers of the mind--accuracy, quickness, memory,observation, concentration, judgment, reasoning, etc.--is to exercise thesepowers. The particular content is not important; the important thing isusing the mind.b. We should teach only those things needed in "real-life" situations.Discuss each of the following in terms of mathematics to be studied later.a. "You can't subtract 7 from 5."b. "The longer the numeral, the bigger the number" (based on whole numberexperiences) •c. "The common denominator algorithm for division of6 + 5 = ~) is better than the invert-and-multiplyfractions (1 + 1. = ---.2. + -2 =. 3 . 1 5 3 2 2 10 6 10one (5 + "2 = 5 x 1 =5)'"7. Give some applications (in or out of school) of each of the following.a. Some principle of the decimal numeration system (you identify the principle)b. associativity of additionc. surface aread. a number has many "names"(18

DIDACTICSTEACHING FOR TRANSFER8.Discuss each of the following in terms of negative transfer.confusions be avoided?a. "1 is greater than i since 7 is greaterb. "I measured 6 cm and 10 cm, so the areac.,,1 2 3 4 12 1""2 x "3 = (; x (; = 36 = "3than 2."is 60 cm 2 ."How might thesed. The two exercises: Write 2.25 as a percent ••. Express 2.25% without thepercent sign.'09. (Discussion) There are many models for fractions: part-of-a-whole, number line,set model, quotient model (see the commentary to FRACTIONS: Concepts in NumberSense and Arithmetic Skills, Mathematics Resource Project). Which of thesemodels are covered in your text series?10. "Knowledge acquired largely by memorization, by the 'pouring-in' process, andwithout many relationships being established with the individual's existingknowledge has low transferability." [Henderson and Pingry, 1953, p. 245] It islikely that you agree with the quote. Review one of your students' most troublesometopics to see how "relationships with the individual's existing knowledge"can be est~blished.11. If we are to teach two related tasks, one easy and one difficult, most of uswould probably feel that we should teach the easy one first. It may be thatteaching the difficult one first would give greater transfer to the easier task.Ellis [1965] notes that the existing research on this easy-difficult vs. difficult-easyquestion does not give much direction, partly because "difficulty" isnot well-defined and methods of teaching a topic can vary. In each of thefollowing pairs, which would you choose to teach first? Explain.a. the volume of a rectangular solid or the volume of a cylinderb. the area of a trapezoidal region or the area of a parallelogram regionc. fraction addition or fraction multiplicationd. decimal arithmetic or fraction arithmetic12. Ellis [1965] reports this surprising finding: For tasks not depending on specificsfrom the original learning, " • • • transfer of training remains roughlyoonstant with varying intervals of time elapsing between the original and thetransfer task" [po 39], in contrast with the usual decrease in retention! Thatis, although as time passes we tend to forget a specific learning like the formulafor the area of a trapezoid, whatever ability we have to use the technique ofderiving such a formula does not deteriorate so much with time. What implicationmight this have for whether we should pay attention to transfer?13. "One seventh-grade teacher tells the class what lies ahead in mathematics bydescribing algebra as mysterious and difficult. 'Instead of working with numbers,you do problems like this'-- ••• an equation (is written) on the board withoutexplaining it. A second teacher does not mention algebra by name, but casuallyuses various formulas such as A = lw and i = Prt when the class is doing numericalproblems on area or investment. Each formula is treated as a shorthandsummary of what the group has been expressing in words. What effect would eachof these methods have upon readiness for algebra?" [Cronbach, 1963, pp. 322-323]How might individual students react in different ways?19

DIDACTICSTEACHING FOR TRANSFER14. It might be worthwhile to skim the units in Mathematics in Science and Society(~with an eye to bringing in certain activities as applications of particular 'mathematical topics.References and Further ReadingsBiehler, Robert. Psychology Applied to Teaching. Boston: Houghton Mifflin, 1971.Bruner, Jerome. Thefrocess of Education. New York: Vintage Books, 1963.Cronbach, Lee. Educational Psychology. 2nd ed. New York: Harcourt, Brace &World, 1963.Ellis, Henry. The Transfer of Learning. New York: Macmillan, 1965.Gagne, Robert. The Conditions of Learning. 2nd ed. New York: Holt, Rinehart andWinston, 1970.Henderson, Kenneth and Pingry, Robert. "Problem-solving in mathematics." TheLearning of Mathematics, Its Theory and Practice. Twenty-first Yearbook of TheNational Council of Teachers of Mathematics. Washington, D.C.: The NationalCouncil of Teachers of Mathematics, 1953, pp. 228-270.Johnson, Donovan and Rising, Gerald. Guidelines for Teaching Mathematics. 2nd ed.Belmont, California: Wadsworth, 1972.Rosskopf, Myron. "Transfer of training."and )?ractice. Twenty-·first YearbookMathematics. Washington, D.C.: Thetics, 1953, pp. 205-227.The Learning of Mathematics, Its Theoryof The National Council of Teachers ofNational Council of Teachers of Mathema-·Shoecraft, faul. "The effects of prov~s~ons for imagery through materials and drawingson translating algebra problems ,grades' sevEm·and nine. ',' C!Jniversity ofMichigan, 1971), Dissertation Abstracts International, Vol. 32A (January, 1972),pp. 3874-5.The use of concrete materials gave the best performance on transfer tasks.Shulman, Lee. "Psychology and mathematics education." Mathematics Education.Sixty-ninth Yearbook, Part I of the National Society for the Study of Education.Edited by Begle, Edward. Chicago: National Society for the Study ofEducation, 1970, pp. 23-71."Youth favor U.S. music, study shows." NAEP Newsletter, Vol. 7, No.5 (September/October, 1974), pp. 1-2.(20

WHATIS A "PROBLEM"?Because of the different uses of the word "problem," it is necessary to clarifywhat the word means as it is used in this section and in the references at the end.A student has a problem whenever (a) thestudent seeks a solution but (b)thereis no obvious way to reach the solution.In particular, a problem situation is differentfrom those encountered and solvedbefore and is not obviously amenable toany algorithm the student has at hand.Hence, whether a situation. is a problemdepends on the student's background.lems.For most mtddle schoolers, theseare. exercises rather than ptoblems:PROBLEMnA.B.STUDENT WANTSBUTSOLUTION,HAS NO CLEAR ROUTE TOSOLUTION.Solving 14 x 8 = n is not a problem for mostseventh graders, whereas it would have been when they were second graders.EXERCISE--YIELDS TOFor aseventh grader, solving 14 x 8 = n involves a routine application of a much-practicedalgortthm; the term "exerctse" is usuallyused to distinguish such types from prob-1. 723 -5}7 = n 2. A frog ate 23 flies.Then it ate 156 more.How many flies did iteat all together?AVAILABLE ALGORITHMThe following could be problems for ffiiddle schoolers (and other people). In eachproblem different letters stand for different dtgtts. (Answers are given at the endof this section after the references.)Problem 1PQ+4773What is Q?Problem 211M- 83NWhat is N?Problem 3 Problem 4 (each digitmust be used)NP.ABxNQ x CPQ to get .DENPthen +.FG22Qto get H.IJWhat is Q? Find each letter's value.Since problems by definition are not solved immediately by tried-and-true methods,we should not expect students to solve them quickly or even to solve all of them.21

DIDACTICSTEACHING VIA PROBLEM SOLVINGAccordingly, it may be better not to assign several problems to be solved within oneclass period. Also, in recognition of individual differences we could include optionsat different levels of difficulty (e.g., "By next Wednesday see whether you can solveone of Problems 1, 2 or 3. These take some real thinking, so don't give up too soon").PROBLEM SOLVING CAN BE THE STAR OF THE CURRICULUMDrill through problem solving.Exercises(in the sense above) are routinelyused to provide practice on skills, butproblem-solving situations can alsoprovide lots of practice--plus othersorts of thinking.If you just skimmedProblems 3 and 4 above, you might tryone of them.Problem 4 will likelyinvolve as much .computation as you wouldordinarily assign. As another example,the solution of Problem 5 to the rightinvolves work with exponents and division(it is not as formidable as itlooks and is discussed further on pages4 and 5). Word problems also supplypractice in solving mathematical sentencesand in computing....------------------,DRILLNEW IDEAS ORPROBLEM TYPES!?roblem 55 999 + 7 givesPROBLEM-SOLVINGPROCESSESPROBLEMSOLVINGwhat remainder?TRANSFERMOTIVATION(New ideas/problem types through problem solving.Suppose the students havedealt only with positive whole number exponents and have just finished a discoverylesson in which the problem was to find a sbortcut for simplifying expressions like57 7 12 10 4 79 71~. They would then be primed for exercises like --- = n, ~ = n, --yr = nand5 7 12 10 0 79 0 0the question, "What seems to be a reasonable meaning to give to 7, 10, 79 ?" andthereby be introduced to zero exponents.Trimble has commented that a "problem is• • . a means to an end; the end is student involvement in the creation or re-creationof the key ideas and techniques of mathematics." [1966, p. 8] Quite a challenge!(22

DIDACTICSTEACHING VIA PROBLEM SOLVINGIf you do, Problem 5 may be only an exercise for you.Imitating solutions to similarproblems is one of our most valuable methods of devising a plan, since it canturn a problem into an exercise. "Can you solve a simpler problem?" The size ofthe exponent causes the difficulty here; 5 998 or 510 or 52 or even 51 (or 5°) wouldbe simpler. But what good would looking at those do? "Look for a pattern!"51 = 5; 5 t 7 gives remainder 552 25; 25 • 7 gives remainder 4 (Is a clear pattern forming already?)53 = 125; 125 • 7 gives remainder 6 (I guess not.)54 = 625; 625 • 7 gives remainder 255 = 3125; 3125 + 7 gives remainder 3 (No one promised there would be apattern! The processes do notguarantee success.)Rut one valuable lesson is to learn topersevere. Students often give up veryExponent Remainderquickly if the answer, or how to getthe answer, is not apparent at once.1 5If we do persist, we would obtain the2 4table to the right, in which the pattern3 6of remainders 5, 4, 6, 2, 3, 1 appears4 2to be repeating. We now have a new5 3problem: using the pattern to predict6 1the remainder when the exponent is 999.7 5We'll leave the rest to you (the answer8 4is at the end of this section).9 6The devise-a-plan hint, "Solve part10 ?of the problem," for middle schoolers·applies most easily to multi-step problems,but is also applicable in many· .· .others. The "Guess and check" technique999 ?1will be mentioned below in connectionwith writing mathematical sentences.Carrying out the plan is usually a pleasure if the plan has been well chosen.Students will, of course, have to keep the plan in mind: remembering what calculationsand what table, deciding what organization, checking patterns, checking computationsfor reasonableness (in particular, if a calculator is used!), and especially keepingthe original problem in mind.25

DIDACTICSTEACHING VIA PROBLEM SOLVINGLooking back is often neglected. Certainly students should ask themselves,"Is this answer reasonable?," and apply some combination of common sense (the girlrode her bike 200 km/h!?) and/or number sense and mental arithmetic. Perhaps mostimportant in the looking back phase is recognizing the problem-solving process weused; for example, in Problem 5 we looked at several simpler "problems" to generatea pattern which enabled us to predict the solution. We could also try to think ofsimilar problems which we think we could now solve. This requires some care (doesit matter that all of 5, 7 and 999 were odd? that 5 and 7 are primes?) • Many problemswill not suggest very imaginative ideas to middle schoolers, but asking theqtl~stiondoes remind the students that PERHAPS THINGS OTHER THAN ANSWERS ARE ALSOIMPORTANT. Finally, looking for another solution will emphasize to the studentsthat quite often there is more than one way to solve a problem.Another example. Several illustrations of problem solving processes with geometryproblems are given in the teaching emphasis ProbZem SoZving, in GEOMETRY &VISUALIZATION, Mathematics Resource Project. Let us look at one more numericalproblem here.(\Problem 6Each different letter represents a different digit.scrAPPAcrSx P and x SAAROOAARt'lN Find each letter's value.(Understanding the problem. We want to multiply a four~digit number (ScrAP) by itslast digit (P) to get a 5-digit number (AARON), which is the same number we wouldget if we reversed the digits of the original number (PAOS) and multiplied it by thenew final digit (S). Different letters stand for different digits, and we want tofind their values. (That was a mouthful, but it illustrates the power of symbolismin mathematics!)26Devising and carrying out a plan.With so many letters in the problem, perhaps wemight first try "Solve part of the problem." Which part? Since the multiplicationalgorithm starts with the units' digits, we might look there. Aha! (?) The lastdigit of P x P is the same as the last digit of S x S--the digit N.now help to narrow the possibilities.Looking at the last digits •Number skills1 x I = 1 4 x 4 = 16 7 x 7 = 492 x 2=!!. 5 x 5 = 2;? 8 x 8 643 x 3 = 2- 6 x 6 = 36 9 x 9 = 81 (gives the following possibilities for P and S:I and 9 or 2 and 8 or 3 and 7 or 4 and 6.

DIDACTICSTEACHING VIA PROBLEM SOLVINGNow what? 8ince we have some information about P and S,we might pursue them.The given data tell us that when we multiply 8 x Pand possibly add it to any digit carried from P x 0 or8 x A, we get AA, a double digit number different fromSOAPx PAARtltJeither PP or 88. 8ubstituting the possibilities generated for P and 8:Similarly,(1 x 9) +(2 x 8) += AA:= AA:(1 x 9) + 1 = 11, which can be ruled out sinceP or 8 =1, so A cannot = 1 also.(3 x 7) + 1 = 22 and (4 x 6) + .2. =12.(2 x 8) + ~= 12 (Why can this be ruled out?)AARGI\lThis last possibility can be eliminated (in a one-digit-times-one-digit calculation,what is the greatest number that might be carried?).P = 7, 8 = 3, A = 2 or P = 3, 8 = 7, A = 2.and3 2x2 277980, we haveThe former does not work (try it, keeping in mind that each different letter representsa different digit).8 1'1 A P7 2 3x 32 2 9AARI'1NTherefore,8ince in the first case 3 x 2 = 6, /J must = 6. The rest is easy . . .7623 3267x 3 and x 722869 22869Looking back. We should be sure to check that our result does indeed give correctcalculations and assign different digits to different letters. We might review howwe proceeded in this problem: We tried to solve part of the problem and were able tobuild on the possibilities generated to get a solution; we obtained information byconsidering units digits of products and the possible "carries" in the multiplicationalgorithm; we ruled out some possibilities by the conditions of the problem and othersby trial and error. Anyone of thesemight come in handy in a similar problem.As another form of looking back,BATH 51.\2.8 MAP~ Ll.~ 7 X P3799G AARONwe might try to make up a similar problem.Or we might ask ourselves, Isthere another method of solution?27

DIDACTICSTEACHING VIA PROBLEM SOLVINGSUMMARY (Even when we distinguish between exercises and problems, problem solving couldserve as the heart of teaching.Problems can pump life intodrill, serve as a vehicle for introducing new ideas and typesof problems, give a basis for transfer, motivate at least somestudents, and above all, put the teaching of problem solving processes into its propersetting--the solving of problems.ODDS AND ENDS ABOUT PROBLEM SOLVINGThis subsection contains a varietyof suggestions and research findingsoriented to the two aspects of a problem:(a) the student must want asolution, but (b)the student must overcomesome block to the solution.The student wants a solution.Let'sacknowledge that many students work problemsonly because that is what is expectedin mathematics classes.Perhaps someof the following will create an interestin the problems themselves.PROBLE."" :STUDEt-lT WAt-lTS SOLUTIOtJBUT OOES t-lOT RE:ALIZE HOWTO REACH IT.Ol-l,OH'IVE GOT APROBLEM.DESIREDSOLU,IOIU• We might poll the class to see what their interests are ("Give 5 things youlike to do," "What are your favorite pastimes--foods, hobbies, TV shows, singinggroups, possessions?") and then try to build problem situations on theseinterests. Listen to the students' topics of conversation in the hallways orbefore class.• Ask each student to submit one mathematics "problem" each week, based on inclasstopics or out-of-school events, and use some of these.• Some teachers find that having the students guess an answer leads to greaterinterest since they then have made a definite "conunitment."• On occasion, a problem can be dramatized by class members (e. g., two "interviewers"questioning a factory owner about pollution control, or two coachestrying to choose the "best" team from given statistics).• Using students' names in problems usually gains some interest if not over-done.Basing problems on things unique to each student (e.g., MuscleFatigue in MATHEMATICS AND BIOLOGY) also "personalizes" the topic.• Problems taken from foreign textbooks or old U.S.to any, may provoke interest.textbooks, if youhave accesc28

DIDACTICSTEACHING VIA PROBLEM SOLVING• Problems based on things close to the students' world can be interesting,although Johnson and Rising offer this opinion: "For most students it is notnecessary to take problems from their immediate environment. Often, studentsare less interested in grocery bills than in cannibals and missionaries, lessinterested in the volumes of oil tanks than in walks through K15nigsberg."[1972, p. 240]• Familiarity may improve the chances of success for some students. For example,Lyda and Church [1964] found that lower ability students in particular were moresuccessful on word problems dealing with situations they were familiar with.Help to overcome the block: general remarks. The major reason for givingattention to the problem-solving processes is that these may make us more selfsufficient.Students need to become more independent and more confident of theirability to do things for themselves. Other people won't always be around! Recentresearch has been investigating systematic instruction in problem-solving processes(e.g., Vos, 1973). Nelson's work [1975] indicated that teaching the use of diagramsimproves problem-solving performance (he also advocates presenting some problems indiagram form).Students with some background in problem-solving processes can beencouraged" to review their own use of the processes ("What approach have Itried?What haven't I tried?"). In any case, attention to processes may help to change theusual student's view that the only concern is The Answer.Middle schoolers may not be confident problem solvers.Having hints in mindis a good idea, although knowing when the students are claiming frustration for thesake of getting a hint and when they are genuinely stuck is a bit of an art;study with ninth graders [Akers, 1975] indicated that having answers accessibletended to increase the time students spent on problems (exercises?); thus, in situationswhere the answer does not reveal how the problems are solved, having answersavailable may lead to greater persistence on the part of the students.OneHaving smallgroups attack problems offers a chance for the students to learn others' approachesand may build up some degree of confidence.Groups usually are more successful onproblems, although there does not seem to be a great carry-over to later, individualwork. lHudgins, 1960] Blomstedt [1974] found that having middle schoolers agree onapproaches to word problems by consensus did seem to enhance their performance onword problems, perhaps because of the immediate feedback and reinforcement fromclassmates."Real" problems will not be solved quickly.for some middle schoolers to accept.time for attempted solutions.This is perhaps the hardest lessonBe sure to point this out, and allow plenty ofA phenomenon you may have noticed in your own problem29

DIDACTICSTEACHING VIA PROBLEM SOLVINGsolving is that after concentrated but unsuccessful work, leaving a problem for aperiod of time often results in a spontaneous or a fairly rapid solution when youreturn to the problem.Hence, you mightmention a problem on Monday, let thestudents work a while, bring up theproblem on Tuesday, perhaps adding ahint or allowing students to look at"hint" cards on your desk (checkingprivately with students who think theyhave solved the problem), bring it upagain on Thursday to check progress(and to remind students of it), andfinally have students show solutions onFriday.(There is no magic in the daysmentioned; however, at some time theunsuccessful students should at least seeYOU M 1G.1-\l TRV50ME SIMPLERPROBLEM':>, AI\lDTHEI\l MAKE ATABLE.someone's solution, if for no other reason than to study how the problem was attacked.)If a student has worked several minutes and is stumped, suggest that he "sleep on it" (and do something else right now.Some work with college students has suggested that females may do more poorlythan males on problems--particularly problems dealing with "masculine" situations-­because girls feel that they are not supposed to do well in mathematics.It is notclear when this particular attitude develops or whether it is prevalent among middleschool girls, but you might be watching for students who seem to have adopted suchan attitude.(Be certain that nothing in your teaching is promoting this attitude!)If there are no female mathematics teachers in your school, you might arrange a guestpresentation by a female teacher from another school or a university, or by somemathematically successful high school girls.Similar success-models could be soughtfor any other identifiable group who seem convinced that they cannot do well inmathematics.Help to overcome the block: word problems. Or are they word exercises?Probably both. We can help to retain their problem character by at least makingsure the students have to read the problem to get a solution.Assignments shouldcontain problems which have extraneous or insufficient data and which require a differentoperation for at least some problem(s) in the list. Otherwise, we may be in (the situation that the Luchinses described more than 25 years ago.C30

DIDACTlCSTEACHING VIA PROBLEM SOLVING(The students) were accustomed to being taught a method and then practicingit; to have to discover procedures was not only quite foreignto them in arithmetic but also in most school subJ'ects. It seems tous that the methods of teaching to which they had been subJ'ected tendedto develop, not adaptive responses, but fixations, so that a childmight know methods and formulas and yet not know where to apply themor how to determine what method best suited a particular problem.Our schools maybe JonJentrating so muJh on having the Jhild masterthe habits, that the hahitsare mastering the Jhild. ILuchins andLuchins, 1950, p. 286, emphasis added]The Luchinses' research has indicated that it is very easy to get a "mental set" in aseries of problems, with the result that one almost blindly proceeds with each newproblem just as he did with the previous problem (see number 10 on page 17).TRAf\JSLATI NGWORDS ----Po. EQUA,\ONEQUATIOt-.)--•• WORDSUsingonly one "type" of word problem in an assignment can foster this uncritical approach,of course, and makes it easy for students to ignore any work with problem-solving strategies.A key question is, how "mixed up" should the lists be? Sumagaysay [1972] hasshed some first light; a sequence of assignments in which the number of types of problemsgradually increased from one type in the first assignment to three types in thelater assignments was superior to sequences in which either only one type was usedthroughout or three types were used throughout (caution: these results cannot beregarded as definitive since the students were likely accustomed to only single-typelists). Here is one final note on mental sets: tension apparently:increases thelikelihood of adopting a mental set.Hence, "'hatever we can do to lessen tension--lessemphasis on time, partial credit, reduced emphasis on grades--may enable students toresist the tendency to fix on one approach.A technique endorsed by many teachersfor word problems is to give lots ofattention to translating from the wordform to the equation (or inequation) formand vice versa.This might include oraldictation of phrases or statements to beexpressed more symbolically ("six more than eight" or "after Don found the five dollarsand twenty-eight cents, he had eight dollars and nineteen cents"),The reverse,giving an equation and asking the students to make up a word problem which fits theequation, can be a valuable diagnostic tool.For example, a student may write aproblem that does not fit a given sentence at all ("I had $11 in the bank at 5%.How much did I get?" for the equation .05 x n = 11), Another diagnostic tool is to31

DIDACTICSTEACHING VIA PROBLEM SOLVING32have students paraphrase a word problem (as mentioned before, paraphrasing also helps (to force an "active" reading as opposed to a recitation of words).Nelson's workcited above suggests that having students write mathematical sentences for given diagrams,pictures or graphs might also be worthwhile.Robertson [1975] has describedsome translation work with balance set-ups, as in the diagrams below.S Ga·· H... c: ...f' = (2. xm) ... 3~QTRAlJ5LATIOl\l :..ITRAI\lSLATIOI0 : t. dU2 TRI\I0SLATIO~: Yl>l'cLgTELL-THE-ACTIONWANTED-GIVENGUESS-AND-CHECK~There does seem to be evidence that writing equations for word problems is aproductive approach (e.g., see Lerch and Hamilton, 1966).how to teach students to write theseequations.There are at least threeapproaches to teaching the writing ofequations for word problems: (a) thesentence-that-tells-the-action approach,(b) the wanted-given method, and (c) theguess-and-check technique.approaches.What is not so clear isWAYS TO CHOOSE EQUATIONS:Let's use the following word problem to illustrate theJan bought a ball glove. Then she decided to buy a ballfor $2.98. In all she spent $16.93. How much did theglove cost?A student who is using the sentence-that-tells-the-action approach is to imagine thestory (bought glove, then bought ball for $2.98, paid $16.93) and write the mathematicalsentence that reflects the action of the story (n + 2.98 = 16.93).Althoughthe resulting equation may not itself indicate what calculation to perform (as inthis example), the method does give an equation which is an accurate "portrayal" ofwhat happened in the problem. [Hartung, 1959] Under the wanted-given approach, thestudent is to determine what is wanted (the cost of the glove) and what is, given (thecost of the ball and the total cost) and then to decide how to get the "wanted" fromthe "given" (16.93 - 2.98 = n). Here the calculation necessary to solve the problemis indicated, but the equation does not "describe" what went on in the problem (theequation 16.93 - 2.98 = n implies that something was removed). With the guess-andchecktechnique, the student guesses the answer (e.g., $15) and then checks it ((15 + 2.98 ~ 16.93). If the guess checks, fine; if not, the student substitutes a 'variable for the guess and solves the resulting equation (n + 2.98 = 16.93).(

DIDACTICSTEACHING VIA PROBLEM SOLVINGWhich way is best?The research to date does not dictate a particular approach.Wilson [1967] found that his wanted-given approach was superior to the tell-theactionapproach with fourth graders.Zweng 11968], however, seriously questionedWilson's study, by pointing out the likely nature of the prior problem-solving experienceof the students.She also emphasized that the tell-the-action approach bestfits the desirable goal of having the mathematics accurately reflect the real-worldsetting.Moreover, another major point she made is that the tell-the-action fourthgraders would have been required to spend some time on the (peripheral) learning ofhow to solve sentences like n + 15 =operation which solves the sentence.93, in which the indicated operation is not theThus, it would be unfair to compare their per~formance to that of students who spent all their time on word problems.AMYl.-:+mMISLEADIt.JG.SKETCHWHO IS TALLER?BY HOW MUCH?Crowe 1l975J,with ninth-grade algebra students, found no difference between the guess-and-checkmethod and an "initial variable" approach (choose a letter for the unknown,write an equation relating it to the other numbers).Whatever approach you choose to emphasize, watch for students who cue themselveson particular words or phrases like "in all" in the word problem above.thenWhile keywords and phrases can tell much about what went on in a problem, they are not reliableindicators of what operation to use."In all" usually does describe a joiningtogetheror accumulating situation, but (as abuve) addition may not be the appropriateoperation to perform. Nesher and Teubal [1975] found that even primary schoOl studentshave begun to rely unquestioningly on cue words and phrases!Students apparently ~reveryuncritical when processing data inproblems. Sherrill [1973] foundthat inaccurately drawn pictures ordiagrams greatly influenced tenthgradestudents, even when accompaniedby the word statement.On the otherhand, presentation of problems throughaccurately drawn pictures or diagramsgave performances superior to word presentationalone. Ammon [1973] foundthat fourth- and fifth-graders respondedwith more ideas when given a picture.BILL1.2.8""33

DIDACTICSTEACHING VIA PROBLEM SOLVINGpresentation rather than only a wordpresentation.Trueblood [1969] recommends,and some text series occasionallyuse, a nonverbal or minimally-verbalpresentation for some problems (asshown to the right in part of PictureProblems 1 from the Addition and Subtractionsubsection of WHOLE NUMBERSNumber Sense and Arithmetic Skills,Mathematics Resource Project).Keil[1965] produced superior performanceon standardized tests by having sixthgraderswrite and solve word problemsabout a given situation instead ofusing the textbook problems(see number5 on page 16).inHow many feet of string?On a 3-day trip,e smJesman drove243 the first day, 319 the seconddaY,and 186 the third day. How fartotal:To by-pass the reading problem of some students, some teachers present wordproblems orally. McGill and Wiles [1975], however, found that for at least some (fifth-grade students, a more important thing is the vocabulary level.Whether problemswith a high level of vocabulary were or were not read ~louddid not make a difference,whereas having problems with a low level of vocabulary read orally (withthe printed statement available) did give a better performance than having only theprinted statement available.Finally, frequent exposure to word problems, combined with some success, shouldbuild confidence.Some easy problems could be given along with more difficult ones.Having at least one word problem a day may be more fruitful thaI! a biweekly or onechapterordeal.SUMMARY• To seek problem materials and presentations that students will find interesting,we might ••---poll the students and eavesdrop on them,--ask them to submit problems,--get them to guess answers and thus commit themselves,--act out selected problem situations, and--use students' names and student situations in problems.(34

DIDACTICSTEACHING VIA PROBLEM SOLVING• To help students to attack problems, we might •--emphasize and practice problem-solving processes,--provide encouragement, hints, peer support, and adequate time,--work on defeatist attitudes,--do what we can to guard against students' developing mind sets,--give attention to translations among mathematical and English sentences and pictorialand concrete representations,--use an approach which fosters the writing of mathematical sentences, and--nourish our students' abilities to process and present word statements of problems.1. Here is a method of combining drillwith problem solving. [Wirtz, 1975JThe object is to start at a ringednumber and find a path through allthe other numbers, ending at theother ringed number. The path mustgo through adjacent squares. Forexample, starting at 6 we can add5 (adjacent diagonal squares are allright) to get to the 11. Subtractionis also allowed.~ 7 47 If.•I~3 5 1110 2-®3 ~5- r=-U10 '2.®Continuing, we can get a completepath as to the right.Try these after the students have theidea; you do not have to indicate thestarting and stopping points. Eachnumber is to be used once and only once.a..2- .3 @b..2- .tt.8 1.0 .6c. Make up one of these by penciling inthe path first, putting in suitablenumbers and then erasing the path.Use whole numbers, fractions ordecimals..8 1 ./4..7 3 G).'2. .3 .3.9 .2- :7d. Modify the idea to use multiplicationand division.e. Modify the idea to allow any combinationof operations.35

DIDACTICSTEACHING VIA PROBLEM SOLVING2. Look at the word problems in one of your textbooks. (a. How many types are in a typical list of word problems?b. Do the settings for the word problems seem interesting? What are the characteristicsof "interesting" settings?3. Students may bring in problems that you can't solve. Plan how you can handlesuch episodes for maximal teaching value.4. Brainstorm with students and other teachers for ways to help students becomebetter problem solvers.5. Since the Keil [1965] results wereso favorable (seep. 14), you maywish to try the same approach withyour students.a. Make up a page like the one tothe right from FRACTIONS:Mixed Operations in NumberSense and Arithmetic Skills,Mathematics Resource Project.CDtJORdJeSS ~Pcblcmsb. Select several pages fromMATHEMATICS AND ENVIRONMENT touse as bases for the studentsto make up their own problems.(Recall that they are also tosolve the problems.)••(6. As was indicated above, you can find at least verbal support for both thesepositions: i. choose familiar situations as sources of problems.ii. choose "exciting" situations as sources of problems.a. What is your position? (Fence-straddling is all right.)b. What are some "familiar situations" to students in your school community?c. What are some problem settings that would be "exciting" to your students?You may be able to get some ideas by asking your most advanced class towrite some exciting word problems for younger students.7. (Discussion) Students with an attitude of inferiority about mathematics problemsmay tend to work more slowly because of lack of confidence and to give up tooSOOn because of a certainty they can't solve the problem. What might you do forsuch students • • •a. during problem~solving seat work?b. during tests? (Do you base the problem~solving portions of your tests onspeed or on power?)(36

DIDACTICSTEACHING VIA PROBLEM SOLVING8. The size or type of the numbers in some story problems sometimes seems to overwhelmstudents' thinRing. Accordingly, some teachers' suggest that the studentssubstitute small whole ilUmbers (or "easy" fractions or decimals), see how tosolve this new problem; and then'proceedin the same way with the original problem.Try this technique on some Word problems from your textbook to seeho~youlike it.9. Where can the following contribute to problem'-centered instruction?a. questioning (See the Questioning section in the resource Geometry andVisualization. )b. discovery lessons (See the section Goals through Discovery Lessons in theresource Geometty and Visualization.)c. laboratory lessons (See the section Teaching Via Lab Approaches in theresource Mathematics, in Science and Society. 210. Suppose you have only containers (unmarked) with the capacities given in each rowand you want the quantities' given in the last column. How would you get thosequantities?H~eW=ta. 29 unit, 3 unitb. 21 unit. 127 unit, 3 unitc. 14. 163, 25d. 18, 43. 10e. 9, 42, 6f. 20. 59, 4g. 23, 49, 3h. 15, 39, 31. 28, 76, 3j. 18, 48, 4k. 14, 36, 8t. 5, 25, 10m. 3, 65, 2920 units, 100 units9952131201825226o3These items were used by Luchins and Luchins [1950J in their work on mental set.It would not be unusual if you used an unnecessarily complicated procedure onparts g, h, j and k. And even college students approachedt and m in a remarkably"set" way. You may wish to give your students a-i. say. from the list (thestereotyped procedure does not work on i). Tell the students not to erase anyof their work.11. How do you teach students to write an equation for the usual word problem?12. Under the tell-the-action approach. it is not the order of appearance of the datain the word presentation but the action and its order in the actual problem settingthat are to be reflected in the equation. For the problem. "Joe got $3.75for his birthday. Then he had $5.25. How much money did he have before his birthday?"one would write n + 3.75 = 5.25 to parallel the actual situation exactly.Write the equations which best parallel these word problems.a. Kay had $6 when she went shopping. When she got home she had $4.15. Howmuch did she spend?b. Dick spent $2.25 at the camporee. He had $1.45 when he got back. How muchmoney did he take to the camporee?37

DIDACTICSTEACHING VIA PROBLEM SOLVING13. What sort of "action" is involved in a percent problem? Look through a textbook (.for other topics for which the "action" is somewhat abstract.14. Mr. Denson:You:"I've tried assignments with more thanthem. The kids just couldn't do them.with only one type."one type of word problem inSo now I use assignments15. Teacher: "Whether you call problem solving the star of the curriculum or theheart of teaching, fancy (and mixed) metaphors don't make problemsolving any easier. I've found that my students can't do word problemsat all, so I just skip them. The kids can learn how to do wordproblems in high school."You:16. Talton [1973] suggests stressing these reading skills to improve performance onword problems: selecting main ideas, making inferences, constructing sequences,following directions for simple and complex choices, and reading maps and graphs.Plan how you might stress some of those skills. (See also Reading in Mathematiasin the resource Ratio, Proportion and Scaling, Mathematics Resource Project.)17. Schwieger [1974] has listed eight components which he believes to be basic insolving mathematics problems. Examine some problems to see whether his componentswould yield a solution. Here is his list:a. CIassify: to recognize pertinent characteristics and attributes of mathematicalproblems or expressions and to specify the class or classes to whichthey belong.b. Deduce: to relate a set of statements so that acceptance of the statementsand their interrelationships dictates a particular conclusion.c. Estimate: to use available mathematical information to make a judgement ofm;a~ent or of a result of calculation.d. Generate Pattern: to put known or available mathematical data into a systematicarrangement.e. Hypothesize: to recognize or to generate conditional relationships betweenmathematical statements.f. Translate: to substitute for one mathematical form, an equivalent representation.g. Trial and Error: to apply knowledge to a mathematical problem in an unorganized'manner.h. Verify: to apply data to a hypothesis in testing its validity.. Ipp. 38-5'9J(References and Further ReadingsAkers, Betty. "The relationship of given answers to persistence and achievement whenattempting solutions to mathematical problems." (Memphis State University,1975), Dissertation Abstracts International, Vol. 36A (November, 1975), p. 2688.38Ammon, Richard. "An analysis of oral and written responses in developing mathematicalproblems through pictorial and written stimuli." (The Pennsylvania State ( ..University, 1972), Dissertation Abstracts International, Vol. 34A (September,1973), pp. 1056-1057.

DIDACTICSTEACHING VIA PROBLEM SOLVINGBlomstedt, Robert. "The effects of consensus on verbal problem solving in middleschool mathematics." (The University of Texas at Austin, 1974), DissertationAbstracts International, Vol. 35A (November, 1974), pp. 2523-2524.Brownell, William. "Problem solving." The PsYchology of Learning. Part II of theForty-first Yearbook of the National Society for the Study of Education.Chicago: The National Society for the Study of Education, 1942, pp. 415-443.Cooney, Thomas; Davis, Edward; and Henderson, Kenneth. Dynamics of TeachingSecondary School Mathematics. Boston: Houghton Mifflin, 1975.Chapter 10 is entitZed "Teaching Prob lem So lving. "Crowe, Kenneth. "The relative effectiveness ofproblems in algebra into open sentences."Dissertation Abstracts International, Vol.two methods for translating worded(University of Illinois, 1975),36A (November, 1975), p. 2732.Davis, Gary. Psychology of Problem Solving. New York: Basic Books, 1973.Hartung, Maurice.instruction.""Distinguishing between basic and superficial ideas in arithmeticThe Arithmetic Teacher, Vol. 6 (March, 1959), pp. 65-70.Henderson, Kenneth and Pingry, Robert. "Problem-solving in mathematics. h TheLearning of Mathematics: Its Theory and Practice. Twenty-first Yearbook ofThe National Council of Teachers of Mathematics. Washington, D. C.: TheNational Council of Teachers of Mathematics, 1953, pp. 228-270.Hudgins, Bryce.Journal of"Effects of group experience on individual problem solving."Educational Psychology, Vol. 51 (February, 1960), pp. 37-42.Jacobson, Ruth.education. II"A structured method for arithmetic problem solving in specialThe Arithmetic Teacher, Vol. 16 (January, 1969), pp. 25-27.Johnson, Donovan and Rising, Gerald. Guidelines for Teaching Mathematics. 2nd ed.Belmont, California: Wadsworth, 1972.Keil, Gloria. "Writing and solving original problems as a means of improving verbalarithmetic problem solving ability." (Indiana University, 1964), DissertationAbstracts International, Vol. 25, Part I (June, 1965), pp. 7109-7110.Kinsella, John. "Problem Solving." The Teaching of Secondary School Mathematics.Thirty-third Yearbook of the National Council of Tea~herb of Mathematics.Washington, D.C.: The National Council of Teachers of Mathematics, 1970,pp. 241-266.Knifong, J. Dan and Holtan, Boyd. "An analysis of children's written solutions toword problems." Journal for Research in Mathematics Education, Vol. 7 (March,1976), pp. 106-112.The ccuthor analyzed one sixth-grade class' errors on word problems from astandardized test. with this class, nearly half the incorrect solutions couldbe traced to computational errors.39

DIDACTICSTEACHING VIA PROBLEM SOLVINGLeder, Gilah. "Sex differences in mathematics problem appeal as a function ofproblem context." Journal of Educational Research, Vol. 67 (April, 1974),pp. 351-353.(\Lerch, Harold and Hamilton, Helen. "A comparison of ato problem solving with a traditional approach."tics, Vol. 66 (March, 1966), pp. 241-246.structured-equation approachSchool Science and Mathema-Luchins, Abraham and Luchins, Edith.anization in problem solv1'1g."(April, 1950), pp. 279-297."New experimental attempts at preventing mech­The Journal of General Psychology, Vol. 42Lyda, Wesley and Church, Ruby. "Direct, practical arithmetic experience and successin solving realistic verbal I reasoning I problems in arithmetic." Journal ofEducational Research, Vol. 57 (July-August, 1964), pp. 530-533.McGill, Melvin and Wiles, Clyde. "Oral reading and vocabulary level as factors inarithmetic word problem achievement. " Paper presented at the annual. meetingof The National Council of Teachers of Mathematics, April. 26, 1975, at Denver,Colorado.Nelson, Glenn. "The effects of diagram drawing and translation on pupils'mathema~tics problem-solving performance." (The University of Iowa, 1974), Di,ssertationAbstracts International, Vol. 35A (January, 1975), p. 4149.Nesher, Perla and Teubal, Eva. "Verbal cues as an interfering factor in verbal (problem solving." Educational Studies in Mathematics, Vol. 6, No.1 Olarch, .1975), pp. 41-51.Polya, G. How to Solve It. 2nd ed. Princeton, New Jersey: Princeton UniversityPress, 1971.Practice in Problem Solving Skills. Books A, B, C. Dansville, New York: TheInstructor Publications, Inc., c. 1974.These books contain spil'it mastel'S fol' aspects of solving 1iIol'd pl'oblems(e. g., identifying missing info1'mation).Riedesel, C. Alan. "Verbal problem solving: suggestions for improving instruction."The Arithmetic Teacher, Vol. 11 (May, 1964), pp. 312-316.Robertson, Jack. "Preparing kids to solve story problems." Paper presented at theNorthwest Mathematics Conference, October 10, 1975, at Eugene, Oregon.Schwieger, Ruben. "A Component Analysis of Mathematical Problem Solving." UnpublishedPh.D. dissertation, Purdue University, 1974.Sherrill, James. "The effects of different presentationslems upon the achievement of tenth grade students."matics, Vol. 73 (April, 1973), pp. 277-282.of mathematical word prob­School Science and Mathe-Sims, Jacqueline. "Improving problem-solving skills."16 (January, 1969), pp. 17-20.The Arithmetic Teacher, Vol. (Sumagaysay, Lourdes. "The effects of varying practice exercises and relating methods40 of solution in mathematics problem solving." (University of Toronto, 1970),Dissertation Abstracts International, Vol. 32A (June, 1972), p. 6751.

DIDACTICSTEACHING VIA PROBLEM SOLVINGSutherland, John. "An investigation into some aspects of problem solving in arithmetic."British Journal of Educational Psychology, Vol. 12 (February, 1942),pp. 35-46.Talton, Carolyn. "An investigation of selected mental, mathematical, reading, andpersonality assessments as predictors of high achievers in sixth grade mathematicalverbal problem solving." (Northwestern State University of Louisiana,1973), Dissertation Abstracts International, Vol. 34A (September, 1973), pp.1008-9.Trimble, Harold. "Problems as means." The Mathematics Teacher, Vol. 59 (January,1966), pp. 6-8.Trueblood, Cecil. "Promoting problem-solving skills through nonverbal problems."The Arithmetic Teacher, Vol. 16 (January, 1969), pp. 7-9.Unified Science and Mathematics For Elementary Schools (USMES)USMES is a major project devoted to interdisciplinary problem solving lSampleproblems--designing lunch lines for best service, inventing a soft drink).For information, write USMES, Education Development Center, 55 Chapel Street,Newton, Massachusetts 02160.Van Engen, Henry. "Twentieth century mathematics for the elementary school." TheArithmetic Teacher, Vol. 6 (March, 1959), pp. 71-76.Vos, Kenneth. "A comparison study of the effects of three instructional strategieson problem solving behaviors." (University of Minnesota, 1973), DissertationAbstracts International, Vol. 34A (November, 1973), p. 2283.Wickelgren, Wayne. How to Solve Problems. San Francisco: W.H. Freeman, 1974.Wilson, John. "The role of structure in verbal problem solving." The ArithmeticTeacher, Vol. 14 (October, 1967), pp. 486-497.Wirtz, Robert. "An individualized diagnostic/prescriptive approach to learning--atthe problem solving level." Presentation at the Northwest Mathematics Conference,October 11, 1975, at Eugene, Oregon.Zweng, Marilyn. "A reaction to the role of structure in verbal problem solving."The Arithmetic Teacher, Vol. 15 (March, 1968), pp. 251-253.Answers to Problems on Pages 1-21. Q = 6 4. .09 5. 6x2. N 66=.543. Q = 4 + .781.3241

(((

DIDACTICSTEACHING VIA PROBLEM SOLVINGSome types of problems are so important that often we deliberately p'rovide muchpractice so that they become exercises. For example, we do want our students to beable to calculate checkbook balances, gasoline mileage, interest, areas of floors,averages, . . . . What we may wish to avoid, however, is making these situationsexercises from the start ("Here's how to do this kind"). Initial work could wellretain the problem nature of the situations ("How do you think we might try tofigure this out?").Transfer through problem solving. Based on his studies, Brownell [1942] hasnoted that concepts and techniques are likely to be most useful in problem solvingwhen they themselves are learned through the solving of problems. As is noted inthe section Teaching for Transfer, working problems in many areas is one of thetechniques for maximizing transfer. Dealing with the various topics from Mathematicsin Science and Society should enhance a student's view of the applicability andimportance of mathematics.Motivation through problem solving.Many students are stimulated by challengeslike Problems 1-5 above, particularlyif the teacher seems to beenthusiastic and interested in theproblem. But many students are not"turned on" by such problems, unlessby chance they get caught up in theenthusiasm of other students, thegroup competition that might be used,or some other feature which may notbe entirely credited to the problem. Since the apathetic student is often the unsuccessfulstudent, it is perhaps of first importance to work for an optimisticattitude: (a) at first choose problems that look like they can be solved (Problems1 and 2, for example) or that have shown a great deal of appeal to former students,(b) present the problems so that the student's ego is not threatened (see the sectionStudent Setf-Concept in Number Sense and Arithmetic Skills, Mathematics ResourceProject) and (c) be optimistic yourself. [Johnson and Rising, 1972, p. 265]23

MATHEMATICS INSCIENCE AND SOCIETYPlacement Guide for Tabbed DividersTab LabelFollows page:DIDACTiCS 12Teaching for Transfer 12Teaching via Problem Solving 20Teaching via Lab Approaches 42Middle School StudentsS6CLASSROOM MATERIALS 70Mathematics and ASTRONOMY 70ASTRONOMY GLOSSARy 120Mathematics and BIOLOGY 126BIOLOGY GLOSSARy ..•............................... 172Mathematics and ENViRONMENT 174Population 188Food 204Air 210Water 222Land and Resources 230Energy 240ENVIRONMENT GLOSSARY 248Mathematics and MUSIC 254MUSIC GLOSSARY 322Mathematics and PHYSICS 326PHYSICS GLOSSARY , 368Mathematics and SPORTS 370SPORTS GLOSSARY 432ANNOTATED BIBLIOGRAPHY 434SELECTED ANSWERS 460

DIDACTICSstars and certain planets.Anotherday, he might have different groupsworking in different locations ("stations")with Mirror, Mirror on theWall, A Swinging Time, Hang Ten andThe Meeting of North and South (allfrom MATHEMATICS AND PHYSICS).TEACHING VIA LAB APPROACHESa.-----------------'----~4. The station approach may also involve different lessons for the ~ topic,with a planned change from station to station either during the same class period oron a different day. For example, studentsmight all be working with equivalentfractions, but one or two groupsmight be comparing models at "circlefractions stations," another groupexploring at a "Cuisenaire rod station,"and another two groups reviewing at"rope stations,lI using a number':"'linemodel.The Teacher RoleALL RIGHT GROUP::',AT "2>TATIO~1. M£A5UR~AlJO FltJD TH~ GROUPSAVERAGE HEIGI·IT. ATSTATIOt.J 2., THE. AVE:\

DIDACTICSTEACHING VIA LAB APPROACHESIt is possible that in early laboratory lessons you will give much direction and exerciseconsiderable control, perhaps starting with a whole-group, teacher-led activity(e.g., fractions with paper'-folding).Later you might have several small groupsworking on the same lesson (e.g., fraction activities on geoboards with a whole-groupfollow-up.)Eventually, if all bas gone well and it fits the unit's objectives, differentlessons could be used at different stations.slow."The usual guideline is, "StartSee the teaching emphasis Lciboratory Approaches in Geometry and Visualization,Mathematics Resource Project, for more examples and suggestions for laboratoryapproaches.Once the students become accustomed to laboratory lessons, the teacher will bedoing these things:--Visiting groups to see whether they are working or are confused.-Asking questions and clarifying procedures, but not serving as an "answerbook."--Observing students as they work.-Encouraging group interaction.--Allowing students to attack the lessons in their own ways. (Sollje errors willoccur but may be caught by the students or exposed by the equiplljent or thedata. Besides, some "errors" may actually be good but unconventional ideaslI--Helping students connect the manipulative to the mathematics being illustrated.(Sometimes students seem to "learn" only the concrete device and not theconcept. )--Talking with other teachers (including librarians) about how to organizematerials.--Being patient.--Praising students for initiative, independence, cooperation, imagination, ••--Providing follow-up to emphasize an idea, to have groups share their findingsor methods of attack, or perhaps to have the students comment on what theylave learned about working in groups.Truly a new role is necessary for the teacher. In assuming such a roleit should also be clearly understood that the activity oriented mathematicsprogram places more, not less, responsibility on the teacher.In traditional(ly) oriented programs, mosf; curricular decisions aredictatedby the textbook. To a far greater extent, in the activity orient~ed program it is the teacher who must know how and what childPen need tolearn, and when they need to learn it.[Reys and Post, 1973, p. 230]IN PRAISE OF LAB LESSONSResearch support for laboratory lessons with middle schoolers has not beenclear-cut.Studies can be found which are highly supportive of laboratory approaches(e.g., Silbaugh, 1972; Schippert, 1965; McClure, 1972), but studies in which laboratorymethods fare no better (or worse) than conventional instruction can also be45

DIDACTICSTEACHING VIA LAB APPROACHESfound (e.g., Smith, 1974). These results are puzzling since many teachers are convincedthat lab lessons are effective, having found that their students have been"turned on" by such lessons.The lack of uniformity for the meaning of "laboratoryapproach" is no doubt part of the reason for the mixed results.Perhaps a more importantpoint is that the major outcomes from laboratory approaches may not be measuredbythe usual paper-pencil methods that research studies are almost necessarilyconfined to.What are some of these other considerations?(No claim is made thatthese outcomes are unique to laboratoryapproaches.)A first area isthat of process-oriented learning.Inlaboratory lessons it is often naturalto focus on how one goes about learningLABLESSONS GIVE OPPORTUNITIES FOR--PROCESS-ORIENTED LEARNING--LEARNER INDEPENDENCE--GROUP MEMBERSHIP SKILLS--GROWTH OF SELF-CONCEPT--CONCRETE BASES FOR CONCEPTSsince the students may be faced with the problem of collecting relevant data, organizingthem and drawing some conclusion (e.g., see Litter Watch in MATHEMATICS ANDENVIRONMENT).A laboratory lesson may also be a discovery lesson in which studentspractice problem-solving skills such as looking at simpler cases or lookingfor a pattern.Exploration and pursuit of open-ended questions can also beincorporated into laboratory approaches--some say they must be.(Related to this process-of-·learning area is the development of learner indepen-·dence and feelings of growing adequacy.That is, the students can take responsibilityfor their own learning ratherthan wait for the teacher to tell themhow to do the problem. No doubt youhave had students who have learned toowell to pretend they are confused orwho seam ~aspend more time asking youwhether their work is correct than they do working.I-\OW COULD'

DIDACTICSTEACHING VIA LAB APPROACHESSince many laboratory lessons involve small groups,they provide excellent opportunities for the developmentof group membership skills and, in many cases, the growthof a student's self-concept. With the aid of concretematerials and group support, some students may be ableLABAPPROACHE~S,..,.._....CO~CRE1gMATE:RIAL~SMALLGROUPSto experience more success than is usually the case. Errors may be recognized fromthe data or the apparatus and be changed before handing in a report. Every teacherknows the importance of peer group reinforcement to most middle schoolers. Smallgroups can be used to increase cooperation and decrease competition in some classes.. • • the laboratory approach builds on success rather than punishfailure, and -"i-"t~u~s.":e~s---""c~ur=-r=-e=cn,-"t~t~h~e",o",n=· e",s,,--"o::..f Lthem.' ..;l",e",a,"rn=i"-n",gL-':r-"a",t::;h",e",r---,t",h",a",n---,i",g:.;n",o=.r,,,-e[Kidd, Myers and Cilley, 1970, p. 128, emphasis added]Most teachers feel that more meaningful learningcan result when students are actively involved andworking with lots of concrete examples. The use ofconcrete representations for ideas has been promotedin these resources. Let us review some of the majortalking-points for concrete materials.CONCRETE MATERIALS1. Foundations for abstraction. Piaget, althoughnot a learning theorist, feels that natural mentaldevelopment is best fostered by an environment rich inexperiences with concrete objects (see Piaget andProportions in Ratio, Proportion and Scaling, MathematicsResource Project). Similarly, Bruner's idea that thebasic level of representation lies in physical actionhas been touched on in The Teaching of Concepts inGeometry and Visualization, Mathematics Resource Project.Experience suggests that only the top students caneasily handle purely verbal and symbolic explanations.LABAPPROACf

DIDACTICSTEACHING VIA LAB APPROACHES2. Multiple embodiments. The Dienes blocks fornumeration are an example of an embodiment of a mathematicalidea. An embodiment of an idea is a (lessabstract) model for the idea. For middle school purposes,embodiments usually are concrete materials or~=1pictures. Embodiments of fractions could be circularor rectangular regions, number lines, or collections ofblocks. Zoltan Dienes, a mathematician-psychologist12.43who has given extensive attention to concrete materialsin his ~ork with school children, feels that as many different embodiments as possibleshould be spread throughout students' work with an idea so that they see theessence of the idea. (All the embodiments should not be presented at One time, ofcourse.) Working only with circular regions as models for fractions, say, mighthinder a young child's ability to "see" one-half of a collection of gumdrops. Hence,Dienes would endorse our usual practice of using many empodiments for fractions:number lines, oranges, cakes, set models, other regions in addition to circular ones,etc. Note that multiple embodiments should also help students transfer their learningto other situations (see Teaching for Transfer in this resource). Finally, dif- (ferent students may be able to "see" a concept best with different embodiments;hence, multiple embodiments may help with individual differences.3. Concept clarification. Dienes also calls for as many variations as possiblewithin each type of embodiment. Forexample, in using circular regions forfractions, a teacher would use regionsof different sizes, cuts with differentorientations, a variety of numeratorsand denominators, and fractions greaterthan one. Looked at another way, theDOE",> THlS SI-\ow-!i:?~principle suggests varying irrelevant characteristics as much as possible. Activitiesfrom Mathematics in Science and Society can be used for topics like proportions, percent,etc., with these ideas of multiple embodiments and maximum variability in mind.4. Concrete materials = realities. Perhaps one of the strongest arguments forconcrete materials is that they link mathematics to real things. It is very easy formathematics classes to deal only with abstract symbols. Students who have generated,gathered, organized and studied their own data in a variety of real contexts shouldgain some appreciation of the relevance of mathematics.48

DIDACTICSTEACHING VIA LAB APPROACHES5. Some middle schoolers are engineers? In commenting on the problem of choosingappropriate experiences for students, Davis gave this opinion based on theMadison Project's work:... fifth graders ... are "natura! inteUectua!s". . .. By contrast,seventh and eighth graders are not "inteUectuals;" it mightcome c!oser to say they are I~ngineers~t heart. For 7th and 8thgraders, the usua! sohoo! regime of sitting at desks, reading, writing,and reoiting seems to ignore the basio nature of the ohi!d at this age;he wants to move around physioa!!y to do things, to exp!ore, to takeohanoes, to bui!d things, and so on. At this age we prefer to get theohi!dren out of their seats and, where possible to get them out of theo!assroom and eVen to get them out of the sohoo!. [1964, pp. 149-150]Perhaps this is why many teachers have such success with laboratory lessons whichincorporate concrete materials.SMALL GROUPSgames,Small groups can be used in a variety of ways--review, drill,Many times laboratorylessons are organized around small groups of students.The reason is not necessarily a lack of equipment foreach student!There are several .potential benefits.LABAPPROACHES/e--~,---', COfJCRET£I ",MA.ERIALS, I\ I, ,/, ,5MALL GROUPSPerhaps the most desirable feature of small groups is that they maximize theopportunities for students to interact.is neglecting one of the main sources of classroom learning."1975, p. 8] Piaget has· indicated thatinteraction with others is an importantkey to mental development since studentsneed to get the views of others and toclarify their own.Learning to be acontributing member of a group, improvinglistening and communication skills,and even taking the responsibility forcareful treatment of materials are allworthwhile achievements.Small groups may help a class become more cohesive."The teacher who discourages peer interaction[Schmuck and Schmuck,Schmuck and Schmuck say,"Frequent interactions among students allow for possibilities of more cohesivenessto emerge; infrequent interaction generally keeps students from getting highly involvedwith one another." [1975, p. 159] In particular, small groups may be a tool49

DIDACTICSTEACHING VIA LAB APPROACHESthat a teacher can use to integrate an isolate (e.g., a new or an unpopular student) (:Lnto the class.The Schmucks point out, "Acceptance by the peer group •.• increasesa student's self-esteem and facilitates working up to potential." [1975, p. 93] Tofoster a more accepting attitude toward an isolate and to focus on the positive, theyoffer these ideas among others [1975, p. 108]:--Have students interview each other and prepare a biography for the person theyinterview. These could be collected as a "class biography," and used in thelanguage arts class.--Form small groups for classwork. Change these every month.Forming GroupsExactly what is the best way to form groups for laboratory lessons is not clear.Homogeneous-ability groups seem reasonable but are difficult to establish and maylimit the richness of the group's findings.Reys and Post review some of the prosand cons for friendship and heterogeneous-ability grouping schemes:. Friendship grouping can result in an excellen,tlearning experience, as the group attacks theactivity wi'th enthusiasm and candidly discusses theresults. On the other hand, grouping by friendshipcan be unproductive and characterized by idle chatterwhen the group's attention is diverted or interestwanes.HOMOGENEOUS ?HETEROGEt>JEOUS ?FRlEI-lDSHIP ?RAr.JDoM ?Heterogeneous grouping can also be used effectively in the laboratorysetting. The brighter children often assume a leadership role answeringactivity-related questions raised within the group. Such childrenoften learn more from the activity when they are asked to explain the"why" to others. Th,e child of lesser ability will also benefit froman explanation provided by a peer, rather than from thE! teacher. Perhaps'i;he greatest fear of the mixed ability grouping is that brightercRi1dren wi'll dominate the activity, thereby intimidating other groupmembers. . .•The discussion suggests that each grouping scheme has its own advantagesand di'sadvantages. Above aU else, (me should not become dependent upona single method. It is often advisable to change the group structure asnew activities are undertaken. The frequent formation of new and differentgroups can result in student growth in both the academic andsocial domains. I1973, p. 224](Because of the greater chance for a puzzled student to get an explanation fromanother student, Dienes and Golding favor heterogeneous groups formed voluntarily,with teacher "suggestions" restoring heterogeneity if necessary. [1971, pp. 139-140]And Smith says, "•.• groups based on a friendship basis often work very well andproduce more outstanding work than groups formed on other bases." [1959, p. 8]( ..50

DIDACTICSTEACHING VIA LAB APPROACHESIn some classes voluntarily formed groups can easily result in one group (ormore) consisting of only the class "rowdies," in a group much more interested intalking about something besides mathematics, in a group consisting of "I ain't gonnado nothing" students, or in a few friendless students without a group.To l1\inimizethe chances of these occurring, you might make the group assignments, using a sociogral1\for some guidance.To make a sociogram, explain that the class will be doingsome group work soon and ask the students to write their names and to list two orthree students they would like to work with.(Point out that you cannot promiseto l1\ake "perfect" assignl1\ents to groups but that you will do your best.)take a couple of tries to sort out themass of arrows, but the diagram to theright shows a COml1\on way of organizingthe choices.Or you might prefer thesociomatrix iorm (see the figure).s:ociomatrix doesil t t"organize" theAIt maystudents as clearly but it is easierSoCrOGRAMto make if many students are involved.1--------------------------1With either form,you can pick outisolates (e. g., Sid) and watch forclusters that you might like toaccept or avoid (like the rowdies).Just as there seem to be no"official" recommendations for forminggroups, there are also none describingthe best group size, The nature ofv .,;the lesson, the amount of equipmentEllYV Vavailable and the number of stationsDEE:.,; Vplanned all have some bearing on the3 02.3 2. 3 1group size. Sometimes pairs of stu-SOCrOMATRlXdents fit an activity very well. Many teachers prefer that students work in pairssince work can be shared and eachD010SIDBILLCHOOSERBE:TTYvv'v'IIVmember of the pair can "keep busy."VV V.,; VGroups ofthree often seem to result in someone's being left out but would fit laboratory lessonswhich take two students to handle the apparatus and one to record the data.Some teachers like to have four students work at the same location, but in pairs; thisplan maximizes involvement yet makes varying viewpoints readily available.thumb is to have groups of fewer than 6 students.A rule ofLarger groups tend to fragment.51

DIDACTICSTEACHING VIA LAB APPROACHESe(IN CLOSINGEach of these tools--laboratory approaches, concretematerials and small groups--seems to have much to offer.Lessons which use all three can work toward higher levelcontent objectives, socialization concerns or firm basesfor concepts.1. Take the objective of one of your recent (non-laboratory) lessons and think ofpossible laboratory approaches for it.2. (Discussion) Think of possible laboratory approaches for these objectives.Compare ideas.a. The student will be able to set up a proportion for a rate situation.b. The student will be able to solve a proportion.c. Some objective that is of interest to several teachers.3. Plan how you would (a) organize the class for, (b) prepare the students for,and (c) follow up each of these laboratory lessons.a. Watts Up? in MATHEMATICS AND ASTRONOMYb. Muscle Fatigue in MATHEMATICS AND BIOLOGYc.d.People-Counting Rates in MATHEMATICS AND THE ENVIRONMENTI Believe in Music in 'MATHEMATICS AND MUSIC(s. Decay and Half-Life in 'MATHEMATICS AND PHYSICSf. Dropping the Ball in MATHEMATICS AND SPORTS4. (Discussion) Compare notes with other teachers about .••a. how best to make up groups for small-group work.b. the best group size for a mathematics laboratory lesson.c. classroom furniture arrangements for laboratory lessons.d. whether an in-class lab is preferable to a lab in a separate room.e. using more than one laboratory lesson at a time, with different groups workingon different lessons.f. how frequently laboratory lessons can and should be used.5. (Discussion) Ms. Doe: "I tried a lab lesson once and just didn't feel comfortablewith it. The students worked at different rates, soI couldn't explain anything to everyone at once. Therewas a little more noise, but I could live with that. Whatbothered me was that I didn't feel in control of theinstruction."What would you say to Ms. Doe to encourage her to try laboratory lessons again?6. (Discussion) Mr. Denson:Do you have any ideas for"I know what Ms. Doe means, but you can get used tonot being 'center-stage.' I use lab lessons often butam bothered by one thing: some of the students justlet the others do the work and the thinking and thencopy down the final results."Mr. Denson?c52

DIDACTICSTEACHING VIA LAB APPROACHES7. Make a sociogram for one of your classes (see p. 9). Often there are surprises,even for a class a teacher "knows" pretty well.8. (Discussion) List as many embodiments (models) as you can for each of thefollowing.a. decimalsb. percentsc. rectanglesd. integers9,. Dienes has also proposed a "deep-·end theory" as a result of some of his work withstudents. "It has been found that, at least in some cases, it is far better tointroduce the new structure at a more difficult level, relying upon the child todiscover the less complex sections within the whole structure." [Dienes andGolding, 1971, p. 57] "Structure" has a precise meaning in the quote, but itis interesting to consider the "deep-end" idea with respect to some of the usualtopics in the curriculum. Can you think of plausible ways to start at the "deepend"for these cases? Would you start there?a. addition of fractions: deep~end--start with unequal denominatorsb. division of decimals: deep-end--start with decimal divisorsc. coordinate systems: deep-end--start with 3 dimensions10. Results in a couple of research studies [Bisio,1971; Vance and Kieren, 1972] indicated that ingeneral teacher~demonstrationswith concrete materialswere at least as'effective as "hands-on" exper~ience for the students (but both are superior tono manipulatives at all). If you have two comparablesections of the same class, you may wish to tryyour own experiment along these lines.LABAPPROACHESCOtJCRETEMATERI"'L

DIDACTICS12.13.14.TEACHING VIA LAB APPROACHESa((continued)performed better than students without partners. Students and teachers found theexperience worthwhile, with major "pluses" being the sharing of skills and thecooperation. Major concerns were social and personality conflicts in the peersand one partner merely copying the other's work. How would you form pairs of studentsin one of your classes?Would you group students on the basis of their body builds?! Yerkovich [1968]found that. for students grouped by body type (non-technically. skinny, wellbuilt.and fat). the average group gains in grade-level scores ranged from 1.7to 2.0. On the other hand. grouping by IQ. interest or socio-economic statusgave an average group gain.."score of at most 1. 3.When a lesson calls for small groups and the unrestricted generation of manyideas ("divergent" thinking). you might consider forming groups which are homogeneouswith respect to divergent thinking. Torrance 11961] found that withstudents in grades 4~6. groups homogeneous with respect to such thinking led toless social stress (bickering, sarcasm. disorder, ••• ) and to greater enjoymentof the task than heterogeneous groups. On the other hand. some homogeneousgroups are not very productive. and in some cases creative thinking is stimulatedby a moderate amount of social stress. How is divergent thinking measured7(Visit the school counselors.) Is divergent thinking important in mathematics7References and Further ReadingsThe Arithmetic Teacher, Vol. 18 (December. 1971).Bisio. Robert.fractionsAbstractsThis issue focuses on the mathematics labopatopy."Effect of manipulative materials on understanding operations within grade V." (University of California at Berkely. 1970), DissertationInternational, Vol. 32A (August. 1971). p. 833.(Carmody, Lenora. "A theoretical and experimental investigation into the role of concreteand semi-concrete materials in the teaching of elementary school mathematics."(The Ohio State University. 1970), Dissertation Abstracts International,Vol. 3lA (January. 1971), p. 3407.Davidson. Patricia. "An annotated bibliography of suggested manipulative devices."..:;Th",-=ec..:.:A-=t-=i-=t;;:hm",'-=e-=t=i-=c-=.Te=a=c",h"-e,,,r:o.' Vol. 15 (October, 1968). pp. 509-512, 514-524.Davidson. Patricia and Walter. Marion. "A laboratory approach." The Slow Learner inMathematics. Thirty-fifth Yearbook of The National Council of Teachers of Mathematics.Washington, D. C. : The National Council of Teachers 0;1' Mathematics, 1972.pp. 221-281.Davis. James. Group Performance. Reading, Massachusetts: Addison-Wesley, 1969.Davis. Robert. "The Madison Project' s approach to a theory of instruction." Journalof Research in Science Teaching. Vol. 2 (June. 1964), pp. 146-162.Dienes. Z. P. and Golding. E. W. Approach to Modern Mathematics. New York: Herder (and Herder. 1971.54

DIDACTICSTEACHING VIA LAB APPROACHESHudgins, Brycesolving."287-296.and Smith, Louis. "Group structure and productivity in problem­Journal of Educational Psychology, Vol. 57 (October, 1966), pp.Hynes, Mary; Hynes, Michael; Kysilka, Marcella; and Brumbaugh, Douglas. "Mathematicslaboratories: what does research say?" Educational Leadership, Vol. 31 (December,1973), pp. 271-274.After reviewing studies dealing with mathematics laboratories, the authorsconclude, "The results of the mQJ'ority of the studies indicated that the mathematicslaborato~ was superior to other methods in increasing achievement,retention, and transfer."Jackson, Robert and Phillips, Gussie. "Manipulative devices in elementary schoolmathematics." Instructional Aids in Mathematics. Thirty-fourth Yearbook ofThe National Council of Teachers of Mathematics. Washington, D.C.: TheNational Council of Teachers of Mathematics, 1973, pp. 299-344.Johnson, Donovan and Rising, Gerald.Teaching Mathematics. 2nd ed."The mathematical laboratory." Guidelines forBelmont, California: Wadsworth, 1972, pp. 446-471.Kidd, Kenneth; Myers, Shirley; and Cilley, David. The Laboratory Approach to Mathematics.Chicago: Science Research Associates, 1970.Kieren, Thomas. "Activity learning." Review of Educational Research, Vol. 39(October, 1969), pp. 509-522.McClure, Clair. "Effectiveness of mathematics laboratories for eighth graders."(The Ohio State University, 1971), Dissertation Abstracts International, Vol.32B (January, 1972), p. 4078.Prielipp, Ronald. "Partner Learning in Secondary School Mathematics." UnpublishedPh.D. dissertation, University of Oregon, 1975.Purser, Jerry. "The relation of manipulative activities, achievement and retentionin a seventh-grade mathematics class: an exploratory study." (University ofGeorgia, 1973), Dissertation Abstracts International, Vol. 34A (December, 1973),pp. 3255-3256.The use of manipulatives resulted in better performance on units dealing withfractions, decimals, using a ruler, and using a micrometer (grade ? students).Reys, Robert. "Mathematics, multiple embodiment, and elementary teachers." TheArithmetic Teacher, Vol. 19 (October, 1972), pp. 489-493.Reys, Robert and Post, Thomas. The Mathematics Laboratory, Theory to Practice.Boston: Prindle, Weber & Schmidt, 1973.Schippert, Frederick. "A comparative study of two methods of arithmetic instructionin an inner-city junior high school." (Wayne State University, 1964), Disser­.:t.:a.:t",i",o;;n,-·;;A",b.::s.:t.::r.::a.::c.::t.::s-=.In=t.::e:;;.rn",a=t~=-'o",n",a=l, Vo1. 25 (March, 1965), pp. 5162- 3•55

DIDACTICSTEACHING VIA LAB APPROACHESSchmuck, Richard and Schmuck, Patricia. Group Processes in the Classroom. 2nd ed. (Dubuque, Iowa: William C. Brown, 1975.Silbaugh, Charlotte. "A study of the effectiveness of a multiple-activities laboratoryin the teaching of seventh grade mathematics to inner-city students."(The George Washington University, 1972), Dissertation Abstracts International,Vol. 33A (July, 1972), p. 205.Smith, Emma. "The effects of laboratory instruction upon achievement in and attitudetoward mathematics of middle school students." (Indiana University, 1973),Dissertation Abstracts International, Vol. 34A (January, 1974), pp. 3715-6.Smith, Louis. Group Processes in Elementary and SecondarySays to the Teacher Series,No. 19. Washington, D.C.:Teachers of the National Education Association, 1959.Schools. What ResearchAssociation of ClassroomTorrance, E. PauL "Can grouping control social stress in creative activities?"Elementary School Journal, Vol. 62 (December, 1961), pp. 139-145.Vance, James and Kieren, Thomas. "Laboratory settings in mathematics: what doesresearch say to the teacher?" The Arithmetic Teacher, Vol. 18 (December, 1971),pp. 585-589."Mathematics laboratories--more than fun?"Vol. 72 (October, 1972), pp. 617-623.School Science and Mathematics,Wallen, Norman and Travers, Robert. "Analysis and investigation of teaching methods. ''>Handbook of Research on Teaching. Edited by Gage, N.L. Chicago: Rand McNally,1963, pp. 448-505.(Wilkinson, Jack. "A review of research regarding mathematics laboratories." MathematicsLaboratories: Implementation, Research, and Evaluation. Edited byFitzgerald, William and Higgins, Jon. Columbus, Ohio: ERIC Information AnalysisCenter for Science, Mathematics, and Environmental Education, 1974, pp. 35-59.Yerkovich, Raymond. "Somatotypes: a new method in grouping." The Clearing House,Vol. 42 (January, 1968), pp. 278-279.56

DIDACTICSTEACHING VIA PROELEM SOLVINGProblem solving prooesses throughproblem solving. (How else?) Developingone of the lists of problem-solvingapproaches (as to the right) through severalwell-chosen problems over a periodof time is one way to start.However,the prooesses must be emphasized andpraotioed frequently, just as computationalskills must be, if they are to becomematters of habit.Getting the studentsto try the approaches may require a"sell-job."Many middle schoolers wantto know specifically what-calculation-todorather than what-kind-of-thinking-todo.Unfortunately, problem-solving processeslike those in the list do notassure a solution.They do, however,give students some things to try when-­as is the case for "true" problems-­there is no obvious way to proceed.us briefly look at the steps listed tothe right.Understand the problem.LetAt minimumthis step includes knowing what is soughtand what information is given, and helpsto "get started" on the problem.Somestudents will have difficulty reading,...---------------,(STEPSIN PROBLEM SOLVING1. UNDERSTAND THE PROBLEM.2. DEVISE A PLAN.--DO YOU KNOW HOW TO SOLVEA SIMILAR PROBLEM?--SOLVE A SIMPLER PROBLEM--MAKE A DIAGRAM) TABLE)NUMBER LINE) GRAPH--SOLVE PART OF THE PROBLEM--GUESS AND CHECK--LOOK FOR PATTERNS--WRITE AN EQUATION3. CARRY OUT THE PLAN.4. LOOK BACK.--CHECK--WHAT METHOD(S) DID YOUUSE?--CAN YOU DO SIMILARPROBLEMS?--IS THERE ANOTHER SOLUTION?(Adapted from Polya, 1971.)word problems (see Reading in Mathematios in the resource Ratio, Proportion andscaltng); attention to vocabulary and emphasis on multiple, slow-paced readings mayhelp some students.Many teachers like to have their students paraphrase word problems.Paraphrasings at least assure that the students have read them and may helpsome break their "I can' tDevise a plan.understand word problems" block.Most often this step is the bottleneck, so more detail is givenin the list above. Let's look at Problem 5: What is the remainder when 5 999 isdivided by 71 (Some students will learn that the usual algorithm does not apply (directly to expressions like 5 999 .) "Do you know how to solve a similar problem?" \(24

to MATffEMATICS AND ASTRONOMYJust like our ancestors who searched the heavens for clues to good and evilomens, we cannot escape noticing the sky. As we look upward, we see the everchangingmoon moving across the sky night after night. Each day the sun rises andsets. The stars reappear each night, but the winter sky does not look the same asthe summer sky.Do you ever wonder why the sun and moon follow these paths over and over? Whythe star constellations are different each season? Why the lighted portion of themoon changes throughout the month? Why Venus is sometimes a morning star and atother times an evening star? Why winters and summers come and go?Since the beginning of history,people have watched the heavens andtried to explain what they have seen.Mathematics has helped them organizeand interpret their observations.Many important discoveries basedsolely on observation were made evenbefore the invention of the telescope.Copernicus demonstrated that the sunwas the center of the solar system andthat the earth revolved around the sun;Kepler concluded that the orbits ofthe planets were ellipses, not circles;early astronomers accurately described the sun's apparent path among the stars.Like the early astronomers, students can discover much about the universe withoutthe use of sophisticated equipment. The activities in this section ask studentsto make observations, record data and use mathematics to interpret their results.A wide variety of astronomy topics are included. Students can determine thenumber of stars visible in the sky at One time without a telescope; make and read asundial; measure the heating effect of the sun; make a scale drawing of the solarsystem; measure the diameter of the moon; discover Kepler's third law of planetarymotion; examine the orbits of comets; build a solar oven; and measure the distanceto the sun. Since distances in the universe are so great, measurements must be madeindirectly. Several methods are presented in the activities. Scale drawings,charts and graphs are used to help students discover relationships.71

INTRODUCTIONMATHEMATICS AND ASTRONOMYThe Table of Contents gives both the astronomy and mathematics topics as wellas a suggested use for each activity. Since it is assumed that all readers may notbe familiar with astronomy, many lessons contain readiness exercises and backgroundinformation. These activities are classified as teacher idea, teacher page, teacherdirectedactivity or activity card. Although those pages labeled as activity cardsare intended as student activities, you may wish to direct a portion of them.Answers to the exercises are provided, and a method for solution is suggested forsome problems. Some lessons contain additional information for the teacher in blue.A glossary of terms, an annotated bibliography of souFces and a sheet which listsnumerical data for the solar system are also included.This section could be used to supplement a science unit on astronomy or byindividual students expressing an interest in the subject. It is hoped that studentswill become more interested in both astronomy and mathematics as a result of seeingsome astronomy-related applications of mathematics.(\((72

MATHEMATICS AND ASTRONOMYTITLEPAGETOPICMATHTYPECounting the Stars75Counting the number ofstars visible in the skyat one time to the unaidedeyeUsing sampling to determinean approximatecountActivity cardManipulativeWatts Up?76Measuring the heatingeffect of the sunGathering and analyzingdataActivity cardManipulativeTrack Records inSpace?78Investigating the gravitationalpUll of the moonand planetsComputing with fractionsor decimalsWorksheetPhases of the Moon79Understanding the phasesof the moonVisualizing and sketchingviews of a sphereTeacher ideaMeasuring byTriangulation80Using triangulation tomeasure distancesindirectly.Using similar triangles,scale drawings, and proportionsto determinedistancesWorksheetManipulativeIt's a Long Shot82Measuring the distanceto man-made satellitesand the moonUsing similar triangles,scale drawings, and proportionsto determinedistancesWorksheetManipulative,Only the ShadowKnows84Using Eratosthenesmethod to measure thecircumference of theEarthUsing similar trianglesscale drawings, and proportionsto determinedistancesActivity cardManipulativeIs the Sun WithinRange?86Using a homemade rangefinderto measure thedistance to the sunUsing a homemade rangefinderto determinedistancesActivity cardManipulativeIt's a Parallaxto Me89Using parallax to measurethe dis·tances tostarsGathering and analyzing dataUsing Angles and proportionsto determine distanceInterpreting diagramsActivity cardWorksheetManipulativeIt's a Great System94Making a scale drawingof the solar systemUsing circles and angles tomake a scale drawingActivity cardManipulative73

(TITLEPAGETOPICMATHTYPEMaking Waves97Using radar to measuredistances in the solarsystemReading a chart and usingproportions to solvedistance problemsWorksheetA Huge Chunk ofCheese98Using the apparent sizeof the moon to measureits diame"terUsing similar triangles andproportions to measure thewidth of objectsActivity cardManipulativeApparently So102Finding the sun'sapparent width fromeach planetUsing a relationship tocomplete a chartGraphing a relationshipWorksheetThat's a Model ICan Relate To103Describing a scale modelof the solar systemInterpre-ting scale drawingsComparing and computing withlarge numbers and decimalsTeacher ideaSize Up theSituation"Hep" Kepler Didn'tGo Around in .Circles105106Interpreting the size ofthe solar system and theuniverseUsing Kepler I s first lawto find a planet's averagedistance from the sunUnderstanding large numbersComparing successive powersof tenDrawing ellipsesComputing averagesTeacher ideaActivity cardWorksheetManipulative(Kepler's Second Law108Using Kepler's secondlaw to study planetarymotionEstimating and measuring areaActivity cardWorksheetNumber Three110Using Kepler's third lawtQ find the orbitalperiod of a planetInterpreting a graphUsing a relationship -tocomplete a char-tWorl

How many stars are visible to the unaided eye on a moonless night? Hundreds,thousands, millions? Make a guess.Surprising as it may seemit is not too difficult to determine their number by direct count.Materials: On heavy cardboard (10 cm x 20 cm) draw a 30°angle, LA, as shown in the figure. Placenails or straight pins at points A, Band Cso that AB = AC. Be sure the nails areupright.In order to count the visible stars you needto select a location where the horizon is nothidden. Hold angle CAB in front of one eyeso that arc BC lines up with the horizon.The night sky should appear like theinside of a hemisphere. Imagine anarc extending upward from theprojected point B on the horizonto your zenith (the pointdirectly overhead) and asecond arc extended upwardfrom the projected point Con the horizon to thezenith. The part ofthe sky enclosed bythe imaginary arcs isone-twelfth of thehemisphere.r---",,~~~~'L-_--HHIORIZE'lHold the cardboard verystill and count the stars withinthe imaginary arcs. The countis easier to make if you observestars in a portion of the skythat does not contain the MilkyWay. Also, you may wish to hold thecardboard so that arc BC is raised a littleabove the horizon to avoid counting stars in the horizon haze.Repeat the count two more times and average your three results.of stars visible in the entire sky at one time is aboutThe total number75

We all know that the Sun gives off a lot of energy. Wereceive this energy in the form of light and heat. Think ofthe Sun as a giant light bulb 149,600,000 km away. How many50-watt light bulbs would we need at the Sun's distance toproduce the same heating effect as the Sun?IMeasure the heating effect of the Sun on an object.Materials: A strip of copper, 6 cm x 3 cm; 1 Celsius thermometer; 1 pencil ordowel rod the same diameter as the thermometer bulb; 1 candle andmatches.a) Lightly pinch the copper aroundthe pencil or dowel rod as shown.b) Bend the ends of the copper backflat.c) Hold the copper over a lightedcandle until the flat surface isevenly coated with soot.d) When the strip cools,slide it from the rodand under the bulb ofthe thermometer.e) Hold the instrument in the shadefor a couple of minutes. Recordthe temperature. Hold theinstrument in direct sunlight fora few minutes (until maximumtemperature is reached). Howmuch was the temperature change?a.)c)DOWEL RODb)BLACK CARBOIIlISA GOOD A8S0RBER.OF' HEAT E~£R.Q,Y.IIMeasure the heating effect of a light bulb on the object.Materials: 50, 60, 100, 150 watt light bulbs andbrackets, pegboard stand, instrumentfrom previous investigation.Use a paper clip to attach the thermometer tothe pegboard so that the copper strip is onthe same horizontal level as the center of thelight bulb...·.... ·····m···. . . . . .. ... . . . ..., ... , .. .. ..... .. · . . .. . .·. . , . . . . .. . '. . .. "· . . . .. . . . ." . . .. '" ....p!!!!t::(l : : ::: : : ::TI-\E WATTNUMBER. OlJT\{E e.ULBTELLS HOWMUCH D.l~RG\(IS PRODUCW\-----J{,EACH SE:CObJo.1) First establish the total temperature change (from room temperature to new (equilibrium temperature) for each light bulb. Be sure each bulb is placed 20 cm76 IDEA FROM: In Orbit- Probing the Natural World, Level Ill, Intermediate Science Curriculum StudyPermission to use granted by Silver Burdett General Learning Corporation

(COI-l">\\JUE:D)from the strip. You should observe that as the wattage (light energy)increases so does the temperature change. Also the time needed for the strip toreach maximum temperature decreases as the wattage increases.2) Now investigate the relationship betweendistance of ~ light source and temperaturechange. Start by p1acirig a 50-watt bulb5 em from the strip. Take seven morereadings by increasing the distance fromthe bulb to the strip by 5 cm for eachreading. Be sure the strip reaches themaximum temperature and then returns toroom temperature before attempting thenext reading. Graph the results.o__- WATTBULB5 to t5 26 'Z5 so gSDISTANCE. OF LIGHT OOURCE* Use the graph to help you decide at what distance the 50-watt light bulb shouldbe placed so that the temperature change will be the same as the change causedby direct sunlight. See answer from I (e).3) Investigate the relationship between distance and power (wattage) if the sametemperature change is desired. Place a 50-watt bulb 20 cm from the strip andrecord the highest temperature. Double the distance and determine the wattageneeded to produce the same temperature reading. Double the distance again.(80 cm). How many 50-watt bulbs are needed at this distance to produce the samereading as one 50-watt bulb at 20 cm?4) Use the relationship in #3, distance measured in2* and the distance to the Sun (14,960,000,000,000cm) to measure the wattage of the Sun.IDEA FROM: In Orbit, Probing the Natural World, Level Ill, Intermediate Science· Curriculum StudyPermission to use granted by Silver Burdett General Learning Corporation 77

TRACK RECORDS IN SPACE??? ---YOU WILL NEED A 1973 (or newer) EDITIONOF AN ALMANAC.Each planet in our solar system pullsobjects toward it with a differentamount of force. In most cases, thesmaller the planet, the smaller thisforce and the easier it would be tojump away from the surface. The ideaof this activity is to find out whatwould happen if you tried 2 sportsevents while standing on other planets.USE THE INDEX OF THE ALMANAC TO LOCATETHE OLYMPIC GAMES.What is the Olympic Running High Jumprecord:What is the Olympic Pole Vault record:1. If you were on the Moon, you could jump 6 times higher.a. Is the Moon's gravity stronger or weaker than Earth's?b. How high would the Olympic record Running High Jump be on the Moon?2. If you were on Mars, you could jump 2t times higher.a. Is Mars' gravity stronger or weaker than Earth's?b. How high would the Olympic record Pole Vault be on Mars?3. If you were on Venus, you could jump It times higher.a. Is Venus' gravity stronger or weaker than Earth's?b. How high would the Olympic record Running High Jump be on Venus?4. If you were on Neptune, you could jump only t times as high.a. Is Neptune's gravity stronger or weaker than Earth's?b. How high would the Olympic record Pole Vault be on Neptune?5. If you were on Jupiter, you could jump only 1; times as high.a. Is Jupiter's gravity stronger or weaker than Earth's?b. How high would the Olympic record Running High Jump be on Jupiter?78SOURCE: Project R-3Permission to use granted by E. L. Hodges

The moon, our largest satellite and the second brightest object in our sky, hashad much influence on our lives. As we watch it throughout the year, the shape of thelighted portion seen from earth changes nightly, but each lighted shape seems to reoccurperiodically. This periodic change in appearance was used by our ancestors tomeasure intervals of time and is the natural unit on which our calendar month isbased. You are probably familiar with the use of the lunar month by American Indians.They measured the duration of trips and other lengthy events by counting the numberof moons that passed--by which they meant complete lunar cycles--before the event wascompleted.Why does the moon's appearance change?predict the moon's approximate location ineightFigure Ishows the moon inpositions in its 29y,-dayorbit around the earth.is off to the righture and lights halfThe sunof the figofthe earthand the moon at each position asshown.the moon,For each position ofpredict its phase(how it appears to you on earth)and sketch your predictions inthe small boxes just outsideFor each time of day is it possible tothe sky if you know its shape (phase)?WAltII-lG GI BBOUSDF'R.$TQUARTERD /---(}.~ ~.«\',, , ,'~ Wft,XII\lG~ U CRESCEI\lT 4011---o () ~'13~ ~+--.0-'---·-FULL \ " NEW SUI.lS, ~ ,loll\ / RAYS'(), ,,(5/ .-.--DWA~t)-_// Dthe moon's orbit. The waxingGIBBOUSWAtJ\t-JGcrescent has been done for you.DTHIRD CRESCEI\lTI'"IGURE. 1 QUARTER ...HOlDT~~~To check your predictions stand about six8ALL SLIGHTLYfeet from a bright flashlight (the sun) in anABOVE YOUR HEAD SOotherwise darkened room. Hold a tennis ball atYOUR BODY DOES "lOTarms length and turn your body very slowlySlOCK THE: LIGHT.while holding your arm steady. The shapes oflighted portions of the ball that you can seewill correspond to the phases of the moon.FIGURE. 2.79

MEA8UI2\NG BYTf2IAtJGULAlI0NHow can we determine the distance to the Sun or to other heavenly bodies? This mayseem like an impossible task since using a metre stick or pacing off the distance won'twork. Actually the problem isn't too difficult. Surveyors and camera manufacturers usea method of measuring distances indirectly. Their method, called triangulation, involvesangles, similar triangles and proportions.ITo measure the distance across a wide rivera surveyor sets up a base line along thebank and marks the ends with stakes A and B.A transit (an instrument equipped with atelescope and protractor) is placed over Aand stake B is sighted. By rotating thetelescope to sight some object (a tree orrock) across the river, the size of angle Acan be read on the transit. The transit isthen placed over B and in a like mannerangle B is measured.- - -~--~.~/F,~~~~" \, DISTA\.ICE:/ - \ ACROSS TI-lE: ~- __ - - ;1 \\ olVEr> --'-' -- ~ - "- ~,,/ ~-.. _.. '\---=-./< )-', --~;-'\ --/ ... ":-~.-~_·__~L~m:.--- __ J~O_~ft~'· ~~-- -A I 100 i'l\E:TRE: BA'5E: U~E. I BNext the surveyor draws a triangle similar to the one above.Note: The distancefrom a point to aline means the lengthof the perpendicularsegment from thepoint to the line.TIIJ THIS ORAWlt..IGI

Use the surveyors' method,MEASUQING BYTQ(AtJGULA1\ON(COMT'MUEO)triangulation, to solve these problems.1. Two lookout towers, tower A and tower B,are 25 km apart.towers spot a fire.Forest rangers in bothThe ranger in tower A uses a transit tosight tower B and then rotates it tosight the fire.of 50° on the transit.ranger in tower B uses aHe reads an angle sizeSimilarly thetransit tosight tower A and then rotates it tosight the fire.He reads an angle sizeof 55° on the transit. How far is the fire from tower A? fromtower B? __________(Hint: Select an appropriate scale, make a scale drawingand use proportions to find the distances.2. A transit is used to sight a flagpole.The base line is 10 metres long. Theangles formed at the base line in sightingthe flagpole are 90° and 73°. Howfar is the flagpole from the baseline?ACCUR.ATE !;CALEDR.A'WI~G'.> AlIlDMl;;A'.>UREMI;;NT5ARE:IMPORIANT.3. Suppose the base line in the above problemis increased to 20 m.mountain and finds the angle ofelevation to be 45°.If one angleof sight is 90°, what would the otherangle of sight be for the same flagpole?(Hint:Modify the scaleli~,..drawing and use a protractor.)\., 0 ..:-;;:·;Challenge: 4. Mountain climber Mary wishes to ~J).': \ ..:.~~know the height of Thunder Moun- __-l:!~ __'~ / · _."-.-'.·..:.· ..i.J:\~':Ltain.At some distance from thebase of the mountain, shemeasures the angles of elevation*and she finds it to be 40°.then walks 300 m closer to theHow high is11-IE Al,lGLE OF ELEVATIONOF" A TREEc.RQUtvOThunder Mountain?The angle of elevation of a tree.IDEA FROM: Charting the Universe, Book 1, The University of Illinois Astronomy Program 81She*

Angle measurements and scale drawingscan be used to determine the distance tothe moon or man-made earth satellites.For example, an observatory in Minneapolisobserves the satellite Echo at 10:30in the evening. It is 36° from the zenithin a southerly direction. (Zenith refersto the position in the sky directly abovethe observer's head.)At the same time astronomers in NewOrleans observe that Echo appears to be 30°from the zenith in a northerly direction.These two observations and a scaledrawing can be used to find the height ofEcho above the surface of the Earth. (Themethod described below works only forcities along the same north-south line.)To make the scale drawing:1. Draw a circle with radius 5 em torepresent the Earth.2. Mark the equator and draw linesegments to represent the parallelsof latitude for l5°N, 30 0 N, 45~N,60 0 N as shown in figure 1.. , , ,, ,......... \ //"'W'GO' t.I LATITUDE:/,"Jt2"'. '30 0 \3Go---•,~o \ ' MIt.lf.l>:APOUS

Over 2,000 years ago a Greek scientist, Eratosthenes (er-a-tas'-the- nez), measuredthe circumference of the earth. His method involved: 1) measuring the angle betweenthe Sun's rays and the zenith and 2) using parallel lines to find the circumference ofa circle.(1. To find the angle between the Sun's rays and the zenith:a. Mount a drinking straw in clay andplace it straight up on a piece ofcardboard. (The straw makes a 90°angle with the board.)b. Place this straw On a level surfacein sunlight and measure the lengthof the shadow cast by the straw.c. Find the length of the straw andmake a scale drawing of the triangle.The drawing could be done on graphpaper.5~ADOW/d. Use your scale drawing to find theangle between the Sun's rays and thezenith.e. Does the time of day affectthe angle between the Sun'sis the shadow the shortest?what way?the size of the angle? Measure the shadow and findrays and zenith at one hour intervals. At what hourAs the shadow increases, does the angle change? In2. To find the circumference of a circle using parallel lines:TCEt0TE:R10

.. ~(CONT'>-lUe:O)3.In the city of Aswan, Eratosthenes observed that at noon on the longest day of theyear the Sun was directly overhead. It illuminated the bottom of a deep well. Atexactly the same time in Alexandria, the Sun was not directly overhead and caused atall pillar to cast a shadow.measuring the pillar and shadow,Eratosthenes found the anglebetweenlthe Sun and the zenithByto be 75°' Pacers had found thedistance from Alexandria to Aswanto be 790 km. Knowing thatlight rays from the Sun strikingthe Earth seem to be parallel,Eratosthenes was able to calculatethe size of the Earth./ISHADOW·SUtJ 1Il0TOVERH£AD1SUI\)OVER\4t:ADIILOCATE A

,~ THE SUNWITt-UN RANGE? (Can a homemade range finder be used to help measure the distance to the Sun?I. To construct the range finder:II.1. Place a large sturdy cardboard boxupside down on a table. The boxshould be at least 50 cm long.2. Cover the upper surface with cleanpaper.3. Draw a straight line segment alongone of the long sides and label itthe base line.4. At one endpoint draw a line segmentperpendicular to the base line andlabel it the sighting line.5. Place straight pins upright at bothendpoints of the sighting line.6. Construct a sighting bar from apiece of heavy cardboard. Draw aline segment in the center andplace straight pins upright at bothendpoints.Move the range finder along thesighting line so that it is closerto the object. Since the sightingbar no longer lines up with theobject it must also be moved. SeeFigure 2.cb oe.:rECT" I ' ,,,\,I \I \, ,I',,, ',,, ,S!~I4T!lJc.L!lJE8 S L1k1EBASE LINEBefore trying to measure the distance to the Sun with the range finder practice usingit.1.2.Pick out an object on the oppositeside of the classroom. Positionthe range finder so that thesighting line lines up with theobject. Place the sighting bar atthe other end of the base line sothat it lines up with the object.See Figure 1.FI~UR.E 1FIGUR.E 2.,I,I,~"" , I, ,, II I, ,, 'm, ,, I: I,', I. ,I ,rU'-------!-ll :, ,l..:~_._. ~((86 IDEA FROM: In Orbit, Probing the Natural World~ Level III, Intermediate Science Curriculum StudyPermission to use granted by Silver Burdett General Learning Corporation

\S THE SUMWITHIN RANGE?(PACOC 2.) DISTAl\lTOBrECTPlace the range finder along the sightingline at varying distances from theobject and align the sighting bar. Youwill notice that as the range finder ismoved closer to the object, the angleformed by the sighting bar decreases.Similarly as the distance increases theangle increases. See Figure 3.CLO&tl

\g THE SUNWITHIN QANGE?(PM€: 3)(\4. In sighting an object, the angleformed by the sighting bar and thedotted line is the same size asthe angle formed at the object(alternate interior angles formedby two parallel lines and atransversal). See Figure 7.AlJGLt" ARE T\1ESAME SIZE..As the distance to objects increases,the angle between the sighting barand the dotted line becomes smaller.FIGURE 7III. Use the range finder to try to measure the distance to the Sun. (CAUTION: DO NOTLOOK DIRECTLY AT THE SUN.)1. Because of the danger of looking at the Sun directly, you must change your methodof sighting. Instead of lining up the pins, line up the shadows cast by the pins.After you have properly aligned the range finder try to read the distance to the (Sun from the scale. You will notice that the sighting bar appears parallel to the'sighting line. The angle between the sighting bar and the dotted line is toosmall to measure. See Figure 8.2. Since this range finder cannotaccurately measg.re.,:t;he distancetotoo--surrTryaltering" the···device. Investigate the effectof lengthening the base line(use two boxes side by side).How does this affect the greatestdistance you can accuratelymeasure with the range finder?AlJGLE I:'TOO SMALLTO MEAe,URE:FIGURE 8How long a base line do youthink is needed to measure thedistance to the Sun?883. Disappointing as it may seem the Earth is not large enough to provide a longenough base line. Astronomers have had to find another way to measure the dis- (tance to the Sun. In the activities It's A Gpeat System and Making Waves youwill discover a method.IDEA FROM: In Orbit, Probing the Natural World, Level Ill, Intermediate Science Curriculum StudyPermission to use f:lranted by Silver Burdett General Learning Corporation

We can use a base line on the Earth to determine the distance to the Moon. However,the Earth is too small to provide a long enough base line to accurately find the distanceto the Sun. How then do we measure the distances to stars?To obtain an adequate base line, astronomersuse the motion of the Earth. The Earth travels inan orbit around the Sun that is almost circular.The diameter of the orbit provides a base line ofabout 300,000,000 km or 2 astronomical units.Astronomers sight the same star fromopposite sides of the Earth's orbit. Insteadof using the angles at both ends of the baseline, astronomers measure only the anglewhose vertex is at the star. From the anglesize and the length of the base line thestar's distance can be found. ,\ .......2. A.iJ, ,,,,I, ,I,ASTROtllOMERSMEASURE THISAI\JGLE.At first it may seem impossible to measure the angle whose vertex is at the star.Astronomers have devised a method called parallax to help them.I. Understanding parallax shift.A. Hold a pencil at arm's length. Lookat the pencil first with one eye andthen the other (close or cover theother eye). The pencil should appearto shift back and forth against thebackground. Hold the pencil close,then far away. The apparent shift isgreater when the pencil is close andsmaller when it is more distant.B. Use the range finder from"How Far Is the Sun." Onthe range finder, draw anarc of a circle with centerat the vertex pin andwith radius equal to thelength of the sightingline. Hold your headabout 30 cm from the vertexpin. Close yourright eye and sight withyour left across therange finder. Have aclassmate place a pin onthe arc so that it linesup with the vertex pin.SIG.H~LII\JE:FROM RIGHT -f------,/ SIGHT LH0EEYEFROI\I\ LEFT EY£:30 em /t:1;if::===='====~VERi!O.llPI\\\IDEA FROM:Charting the Universe, Book 1, The University of Illinois Astronomy Program89

Repeat with your left eye,being careful not to move yourhead.The difference between the twoarc readings tells the numberof degrees of shift the vertexpin moves as seen first withone eye and then the other.The degree of shift is alsothe angular separation of youreyes as seen from the vertexpin. Repeat the previousactivity for different distancesand record the shiftfor each distance in the table.OISTAiJCE FROMV£12T£X PllIl2.0 em'+0 em80 em..IGO emD£GRl:.E"OF SHIFT0° 00 Look fora pattern.II.C. Some stars show an apparentshift also. When astronomerstake photographs of the skyfrom opposite sides of theEarth's orbit, the nearerstars appear to change positionin relation to the backgroundof more distant stars.These background stars do notappear to shift since they areso far away. Since the astronomerknows the position of thebackground stars very accurately,they serve as the degreemarks for measuring the shift.From this measure the astronomercan determine the distanceto the star.Using the parallax angles to find distance.A. To the right are picturesof the bright star Regulusas observed six months apart.The parallax shift isthe same as the angularsize of the Earth's orbitas seen from the star.I'".,..- ..,.(The space between eachline represents a verYlsmall angle, only 50 000of a degree.'.Two starshave not movednoticeably. c:Regulus has .shifted 1 unit.90IDEA FROM:Charting the Universe, Book 1, The University of Illinois Astronomy Program

, ,,,­B. II,,,,, ,,, ,,EI\12T\-l1The larger angle at Regulus measures ~~~c\of a degree. The base line is the 50,000diameter of the Earth's orbit and measures2 astronomical units.For convenience astronomers call half of theThe base linelbecomes one astronomical unit.Regulus is 100,000 of a degree.angle at Regulus the parallax angle.The parallax angle (or parallax) ofParallax angles are very small. Theangular size of alpenny 4 kilometresaway is about 3600 of a degree. Noparallax so large has ever beenmeasured for a star.I:§G;QOOF' A DEGREEFor smaller units of measure, each degree isdivided into 60 equal parts called minutes, so1 degree = 60 minutesMinutes are also divided into 60 equal partscalled sp.conds, so11 second = 3600 of a degreeo oooThe parallax ofRegulus is about .041or 25 of 1 second.C. To compute the distance of Regulusfrom its parallax (zk of 1 second),imagine a large circle, center atRegulus, that passes through theEarth and Sun. First find thecircumference of this large circle.1The angle 25 of 1 second cuts offan arc of length 1 a.u., so 25 a.u.along the circle must make an angleof 1 second at Regulus.AIJ EXAGG£RATE:DANGL£ OF is OF ISECOtJD.An angle of 1 minute would cut offan arc of length 1500 a.u. (60 x25 a.u.); an angle of 1 degree wouldcut off an arc of length 90,000 a.u. (60 x 1500 a.u.). The total circumferenceof the circle is 32,400,000 a.u. (360 x 1500 a.u.) or 3.24 x 10 7 a.u. To findthe distance from Regulus to Earth, find the radius of the circle: divide32,400,000 a.u. by 2 times pi. The radius is 5,200,000 (5.2 x 10 6 ) a.u. One 14a.u. ""150,000,000 km, so the distance is about 780,000,000,000,000 (7.8 x 10 ) km.IDEA FROM: Charting the Universe, Book 1, The University of Illinois Astronomy Program 91

(4. Because distances to the stars are sounit of distance called a light year.travels in one year.large, astronomers have created aA light year is the distance lightThe speed of light is 300,000 km per second.How far does light travel in one year?Find the distance to Regulus in light years.Exercises:1. a. Complete the tableat the right./' - -", -PARALLAX J\() APPROXIMATS DI5TAf-lCE:Remember: (from STAR~,E:COI\lDS (p) 1/\1 LIGHT- YEA12S (d)I.B) as the distancedoubles the 0.8 "+.5 ALP>-lA CCIVTUARIparallax angle ishalved. A00'4 9 SIRIUS0.2 c ALTAIR.You"'-r answer for0.1 ARCTUR.US, veGARegulus will notagree with the 0.05 ALDE.BARANresult above. 0°Both answers are 0°.0,,+ REGULU5only~approximations. .02 A.NTARES(b. Do you see a relationship between P and d?Note:Distances beyond 500 light years cannot be determined using the parallaxsystem. The parallax shift is so small it cannot be measured accurately.For these distances the diameter of the earth's orbit is too short of abase line.c. The closest star, Proxima Centauri, is about 4% light years away or 4 x 10 13 km.If it were added to the scale model of the solar system described in the activity1nat's a Model I Can Relare To, it would lie 1,800,000 metres away from the Sun-­a distance greater than 25 the circumference of the Earth.(92IDEA FROM:Charting the Universe, Book 1, The University of Illinois Astronomy Program

2. The diagrams below represent two photographs of the same part of the sky takenthrough a telescope six months apart. Each photograph represents only one secondof arc.a. Which is thenearest star?b. Which are themost distantstars?6;-.A0CBFMAY~ AI;:CDe;Fc. What is theapparent shiftof star B?~I SEC.OtJD 01'" ARC--.j~I SCCOI\lD OF AlCC----.Jd. What is the parallax angle of star B?e. What is the parallax angle of star D?f. Star D is 18 light years away. How far is star E?g. How far is star A?IDEA FROM:Charting the Universe, Book 1, The University of Illinois Astronomy Program93

(Astronomers are able to construct a scale model of the solar system without knowing asingle actual distance in the system. The construction is based upon accurate observationsof the positions of the Sun and planets. One of the first men to gather suchobservations was Tycho Brahe (TIE-koeBRAH). He lived from 1546 to 1601. Tycho'sassistant, Johannes Kepler (yoe-HAHN-es KEP-ler) used the information to make a solarsystem model.Why is a model of this type useful? If one actual distance is measured, the scaleof the model can be established and other distances can be determined. For example, bymeasuring the distance from Earth to Venus, and using the model, the distance from theEarth to the Sun can be learned.Kepler's method is described below so you can make a scale model of the solar system.You will need a protractor, compass and two partners.I. The orbits of the inner planets, Mercury and Venus, can be positioned in the scalemodel by finding what is called their maximum angular separation from the Sun. (Inpositioning the inner planets we assume that the Sun is the center of the Solarsystem; that Mercury, Venus and Earth revolve around the Sun in roughly circularorbits; and that Mercury and Venus are closer to the Sun than the Earth.)1.To help you visualize Venus' motionaround the Sun as seen from Earth,choose one partner to represent Venusand the other to represent the Sun.You represent Earth. Have "Venus"walk slowly around the "Sun" as shownin the sketch.You point with one hand at "Venus"and with the other at the "Sun."As "Venus" moves, notice how theangle formed by your arms changes.When is the angle very small? Whendoes it get very large? Does it everget as large as 90°?--- ---- "VElJU5'" ,\I,, I------------ -~~/, ,, ,, ,JE-+----c,\, ,~ \', ,,'(_____ -~"'~':::::::: .:~"~ I'd~lJU'!>From his observations, Tycho Brahe 1""O---~jL--.:-o-r-~~~'concluded thaft the mafximumhangsula,;separation 0 Venus rom t e un 1S ~,~ ~ (47°. 'OE:P8

2. To construct a scale model of the orbits of Earth and Venus:a.b.Draw a circle, radius 5 cm torepresent the orbit of the Earth.Mark the Sun by placing a dot atthe center of the circle. At anypoint on the circle place a dotto represent the Earth. Since wedo not know the distance in kilometresfrom the Sun to the Earth,call the distance one astronomicalunit (1 a.u.) as Kepler did.Draw a line segment from the Earthto the Sun. Use your protractorto draw in the Earth-Venus line (EVfor the largest angle of separationTo draw the orbit of Venus, placethe compass point at the Sun and,by trial and error, adjust thecompass opening so that the circlejust touches the EV line. Make asmall dot at the point of contactand label it Venus. (Geometrically,the location of Venus is on theperpendicular from the Sun to theEV line.)Eline)(4r)..sl:ARTH Ic--l-----'--SUI\Jc. Measure the distance from the Sun to Venus in centimetres.How many astronomical units is Venus from the Sun?11\1 YOUR DRAWIt.lG5 CJYn REPRESEI\JTS. 1 A"',RONOM ICALUt.lIT. f-""-_''E v UI\JE3. The greatest angular separation of Mercury from the Sun is about 23°. Draw theorbit of Mercury in your scale model. Measure the distance from the Sun toMercury in centimetres.How many astronomical units is Mercuryfrom the Sun?II.As seen from Earth the positions of the outer planets (Mars, Jupiter, Saturn, Uranus,Neptune and Pluto) in relation to the Sun are much different than those of Mercuryand Venus. To observe this repeat the activity in I(a). Choose one partner to beMars and the other to be the Sun. Again you represent Earth. Arrange yourselves asshown in the figure. As Mars orbits the Sun, observe how the angle between your armsincreases up to 180° and then starts decreasing.1. Since the outer planets can appear anywherein relation to the Sun, Kepler hadto devise a new method to place Mars andthe other outer planets in the scalemodel./ '-'--11--- 'f," SUfV ;-'.......... _------"'.-a. Draw a circle, radius 5 cm as before,place a dot at the center to representthe Sun, and a dot on the circleto represent the Earth.e,UI.l. JEARTHIDEA FROM:Charting the Universe, Book 1, The University of Illinois Astronomy Program95

. Mars orbits the Sun in 687 days, so it returns to the same position every687 days. The Earth orbits the Sun in 365 days. On the circle where willEarth's position be in 687 days? (Since the Earth makes one revolution-­360 0 --in 365 days, it travels about 1° a day. In 687 days the Earth makes1 revolution and travels 322 days ofthe next trip, or about 322°. Howmany degrees are left to complete acircle? ) Mark a dot onthe circle to represent Earth's newposition.c. By observing the angular separationof Mars from the Sun on day #1, andon day #687, Mars' position in themodel can be marked.On day #1 the angular separation is110°; on day #687 the angular separationis 150° as shown in thefigure. Draw these angles in yourdiagram. The point of intersectionis the location of Mars.~-'-I----1 EARTHAT DAY"IkA12n1 ATDAY "G87(d. Use your compass to draw Mars'orbit. Measure the distance fromthe Sun to Mars in centimetres.How many astronomicalunits is Mars from the Sun?Remember: theEarth is 1 a.u. from the Sun.• o'EDAY "G87( ,2.Jupiter orbits the Sun in 11 years 316 days.At one time the angular separation of Jupiter 0from the Sun is 180°. Eleven years and 316 cdays later the angular separation is 122°.Use this information to position Jupiter inthe model. How many astronomical units isJupiter from the Sun?3. Saturn orbits the Sun in 29 years 165 days.At one time the angular separation of Saturnfrom the Sun is 94°. Twenty-nine years and165 days later the angular separation is 89°.How many astronomical units is Saturn fromthe Sun?

In It's A Long Shot triangulation is used to measure the distance to the Moon.Radar, coined from the words RAdio Qetection ~d ~anging, has also been used to measurethe distance to the Moon. What is radar and how does it work? A radar set sends out aradio signal which will bounce off an object and return to the station a bit weaker.Radio signals travel through space at the speed of light, 300,000 km each second. Bymeasuring the time it takes a radio signal to travel to a target and back, it is possibleto determine the distance a target is from the radar set.Exercises:1) The U.S. Army Signal Corps first sent a radio signal to the Moon in 1946. It took2.5 seconds for the signal to travel out, bounce off the Moon and return.Find the distance to the Moon.2) In 1961 radar waves were first bounced from the surface of Venus. When Venus isclosest to the Earth the round trip time for the signal is about 280 seconds. Howclose is Venus to Earth?3) In the activity It's A Great systemthe distances in astronomical unitsof the planets from the Sun wereestablished. These distances aregiven in the table to the right.Use the result from problem #2 andthe information in the table tohelp you answer these questions.a) In astronomical units how faris Venus from Earth when it isclosest to Earth?b) How far is the Earth from thes~n in.astronomical urtits? Inkilometres?c) Find the distance in kilometresfrom each planet to the Sun.Distance from thePlanets sun in a. u.Mercury .39Venus .72Earth 1Mars 1.52Jupiter 5.20Saturn 9.54Uranus 19.18Neptune 30.06Pluto 39.52Can radar be used to accurately measure the distance to the Sun? So far scientistshave been unsuccessful. Since the Sun is composed mainly of hot gases, it is a softtarget and does not reflect the radio waves. However, by using radar to measuretthedistance to Venus and applying this to the solar system model in It's A Great System wecan determine the distance to the Sun."'1M ArJ 0OLD SOFTIE.00 """IDEA FROM:In Orbit, Probing the Natural World, Level III, Intermediate Science Curriculum StudyPermission to use granted by Silver Burdett General Learning Corporation97

f(ILR HUGS·CHUNK:. OF CH£ESED.("ON.I~J 0 o (HOvJ fllG IS rGov6vP,] 0 o l LETS ""6..J151,.-11:: 1f.leMOONf HON611&IIMOON?.- . ---n _'I SL_ ~/~o,sir?IIABc\JrIW INCHAr-J.D AVe Se+\THG •It~ .. l .rTo Peter the Moon appears to be 1% inches wide.correct?Is his measurementReadiness Exercises:1. Close one eye and hold a penny near theother eye so that it just covers theclassroom clock. Hold the penny closer,then farther away. How does theapparent size of the penny change as itsdistance from the observer changes?(2. To tell someone the apparent size of an object, we give its angular size. Theangular size of an object is th.e measure of the angle formed at the observers eye bythe object. The range finder in Ie the Sun Within Range? can be read to measureangular size.a. Draw a large circle on the chalkboard.Draw in a diameter and label the endpointsX and Y.b. Set the range finder halfway across theroom and line up the sighting line withpoint X.c. Use the length of the sighting line todraw an arc of a circle (center at T)as shown in the diagram.d. Sight point Y from T and place a pin onthe arc to mark the line of sight.e. Measure the angle at T with a protractor., ,,,,~~~~~~-!J M£A'.>URESIGHiIlJG TO FlhlD ,\1£LINE AIJGULAR SIZE(98IDEA FROM:Charting the Universe, Book 1, The University of Illinois Astronomy Program

(PAGE: 2.)f. Repeat steps b through d to measure the angular size of the circle at variousdistances from the chalkboard.How is the angular size of an object affected by its distance from theobserver? For additional practice measure the angular sizes of severalround objects in the room.3. Similar triangles are used to determinethe actual size of an object when itsangular size is known. To help youidentify the similar triangles:a. Draw and cut out a 2 cm and a 4 cmdisk. Mark the center of each disk.b. Draw a 10° angle.c. Place both disks in the 10° angle sothat the disks just touch the sidesof the angle. Measure the distancefrom ~he center of each disk to thevertex of the angle. The ratio ofthe distances is about 2:l--the sameas the ratio of the diameters.BOil-l D15K.S HA\JETHE: SAME" APPARENTOR ANGULAR SIz.eoo'?s~ ~~.~~Q:$==o\f'. VoTRIAtIlGLE: RST ISSIMILAR TO TRIAlI.lGLE:. RUV.RB: RA =UV:STd. Repeat c for a 20° angle.IDEA FROM:Charting the Universe, Book 1. The University of Illinois Astronomy Program99

R HUGE"CHUNKo. OF CHEESE}).(PAGE: '3)(The following investigations use the apparent size of an object to find its actual size.1. Students work in pairs. Place a globeat the front of the room. One studentcloses an eye and holds a pencil uprightin front of the second eye sothat the eraser just covers the globe.The other student measures the distancefrom the eye to the pencil, thedistance to the globe and the width ofthe eraser. Estimate and then try tofind the diameter of the globe from themeas urements.Hint: Make a sketch involving similartriangles; A proportion can be used tofind the diameter of the globe.THE PElUC1L AIUDTl-I£ GLOe,C HIWETHE $AME ANGULARSIZE.diameter of pencildistance to pencil= diameter ~f globedistance to globe2. To measure the diameter of the Moon,a night when the Moon is full. Eachdent will need the help of a friend.semble the equipment pictured to thepickstu­Asright.It-loa CARD("'~" PIIUI-lOL£ TO SIG\1T MOOIU',' -::··..•AI0D PEMCILPoint the metre stick toward the Moon andsight the Moon through the pin hole. Slidethe pencil along the metre stick until theTAPEthickness of the pencil just covers theMoon. Read the distance from the pin holeto the pencil. Make several measurementsand average the distances. Measure thethickness of the pencil. The distance tothe Moon is about 380,000 km. Use these FLASHLIG.HT TO \-lELPmeasurements to find the diameter of the RECORD Dle.TAIUCe:Moon.IDEA FROM:Charting the Universe, Book 1, The University of Illinois Astronomy Program(100

A HVGE"CHUN\

In A Huge Chunk of Cheese studentsobserve that the apparent size of anobject depends on the distance from whichit is seen. From the Earth the Sun appearsto be the same size as the Moon. In fact,during a total solar eclipse the Mooncompletely covers the Sun.THE 1'11001\1 APPEARS TOCOMPLETELV COVERTHE SUN.SUlJEARTH(How wide would the Sun appear if theEarth were half as far from the Sun?Since the distances are so great the Sunwould appear twice as wide. Similarly ifthe Earth were one-third the distancefrom the Sun it would appear three timesas wide.The Sun's apparent width (w) isdefined by the relationship d x w = 1where d represents the observer'sdistance from the Sun.How wide would the Sun appear ifviewed from each of the other planets?From which planet would the Sun appearthe largest? the smallest?Use the relationship above (d x w = 1)to find the apparent width of the Sun fromeach planet. Complete the table. Eachdistance from the Sun is measured inastronomical units (a.u.).(CALC ULAiORSCOULD BE USED.°0 ~DISTAIJC£ OF PLAIIJETFROM 5U~ 11\.1 A.U.APPARE.f\1T WIDTHOF THE SUN.4 .7 1.5 5.2 9.5 19 30STUDIC.WTS COULDGRAPH THERELAT10lJSI~IPd X\iJ ' 1 mR THECLOSER PLAIJET5.a) How many times as wide does the Sun appearfrom Mercury than from Earth?3w102b) How many times as wide does the Sun appearfrom Venus than from Saturn?If the Sun appeared to be this8size from Earth, its apparentsize as seen from Pluto would SlJ Wbe smaller than the period atthe end of this sentence.IDEA FROM:Mathematics A Human EndeavorPermission to use granted by W. H. Freeman and Company Publishersd.I 2 3 q 5 Q 7 e 9 to II 12DISTANCe IN A.U.(

THATS A MODELI CAf\J I2ELATE TOe,ATUR.1\)TUPITER.,';tJEPTUIJ£URMJUS.. .,'MAR.S \I£!JU5 .":'.000 0 .: .MERCURY',: .EARTH :: SU1\1The relative sizes of the planets are shown above.The distances between planets is not accurate.Can a suitable scale be chosento use in making a scale drawing thataccurately repr~sents both thedistances and si~es of the planets?The table to the right lists boththe distances and diameters.A scale drawing that uses thesame scale to represent both thedistances and diameters would not bepractical since the distance numbersare so much larger than the diameters.SUl-JAV£R"'C.E DIe,TANCEFROM SUI\)DIAMEn:121,392,,000 kmMERCURY 5'l9OQOOO km "1,8'18 kmvwus lOB,ZOO,ooolon 1'2,112 k.vnEARTH 149,GOO,OOO km 12.,75G k.vnMAR.e, 227,900,000 km

THATS A MODELI CAN I2.ELATE TO(cot-.>1"l tVue:o)(,\In addition, when they see sketches of our solar system, they should realize thattwo scales are used in the picture: one that gives the relative sizes of the planets;one that gives their relative distances from the Sun.Extensions:1) The data in the table on the previous page could be used for rounding exercises.Round the Earth's diameter to the (a) nearest hundred km (b) nearest thousand km.2) Each distance in the table could be written in scientific notation.3) Questions that compare distances could be asked.(a)(b)How much farther from the Sun is Mars than Earth?Which planet is about 100 times as far from the Sun as Mercury?(c)Is the distance of Neptune from the Sun more or less than four times thedistance of Jupiter from the Sun? How much more or less?4) Information in the table below could be used for drill and practice exercises indecimal computation. A similar table could be developed using the planets'average distances from the Sun.(\PLA~E:TOIAME:TEl2.Find thediameterof eachplanet.EARTH 12, 75G k.mMERCURY .38 , !:.A12T~:" DIAMETE.RVENU

One of two movies, Powers of Tenor Cosmic Zoom, or the book, Cosmic Viewby Kees Boeke, can be used to emphasizethe immense size of the solar system andthe universe. If the book is used, theconcept can be made more relevant byhaving students construct on paper asquare 1.5 metres on a side. In onecorner draw a series of squares 15 em,1.5 em and .15 em on a side. The ratioof these sides will show four successivepowers of ten. The measurement of 15 emis being used because it corresponds tomeasures used in the book.Outside, have students mark a square15 metres on a side and place the papersquare in one corner. If the schoolground is large enough, mark a square150 metres on a side.Use the legend of the map to make ascale drawing that represents a square1500 metres on a side. Position thedrawing on the map so the school is inone corner. By relating the series of squares to the pictures in the book Cosmic Viewnumbered -2 through 4, students might get a "sense of scale." The square .15 em on aside will be similar to the picture numbered -2, and the city map square will be similarto the picture numbered 4.The films are available from:POWERS OF TEN (8 min. color)1968 Producer: Charles EamesThe University of Southern CaliforniaDivision of CinemaFilm Distribution SectionUniversity ParkLos Angeles, CA 90007Rental $10.00COSMIC ZOOM (8 min. color)1970 Producer: National Film Board of CanadaContemporary/McGraw Hill FilmsWestern Regional Ofc.1714 Stockton StreetSan Francisco, CA 94133Rental $12.50105

t~£p'KEPLER- DIDNTGO AeOUNDIN CU2CLESIn studying Tycho Brahe's observations of theplanets, Kepler discovered that the orbits of theplanets were not circles, but ellipses.You can draw ellipses of different sizes andshapes.PLAtJET(Materials:A piece of heavy cardboard (20 cm x 30 cm),2 thumbtacks, a piece of string (40 cm)1. Place the thumbtacks in the cardboardabout 5 cm apart. Knot the ends ofthe string and loop it around thetacks.2. Put the point of your pencil insidethe loop and pull the string taut.Draw a figure by moving your pencilin a clockwise direction, keepingthe string taut at all times. Thefigure you just drew is an ellipse.3. Draw several other ellipses usingthe same loop, but with the tacksdifferent distances apart.mcesFOCI

~~t:P'The actual orbits of most ofhelion each year about January 4July 5 the Earth is at aphelion:KEPLER. DIDNTGO A120UNDIN CI2CLES(COIUTltJUEO)the planets are almost circular. The Earth passes periata distance of 147,100,000 km from the Sun. About152,100,000 km from the Sun.Exercises:1. Complete the table by finding the average distance of the Sun from each planet. Usea calculator to help you.PLAtlJET DISTAIJCl;. AT APf1EUOI0 DISTAMCt. AI PERIHELIa},)AVERACE D\S,AlJCEFROM ,HE SUI0a MERCURY G9, BOO, 000 km 46, 000, 000 k..m '" VENUS 109, 000, 000 km 10'7,400,000 km~'HE ORBIT5 EARTH 152.,100, 000 k..m \tt'7, 100,000 km.OF MERCURYAWD PLUTOMARS 2/+9, IDa, 000 k..m 2.0G, '700, 000 km'> ARE THl2. ,jUPITER 815, 000,000 km '7'-\.1,000,000 kmL£ASTSATURN 1,50,,000,000 km 1,3";-" 000, 000 k.mCIRCULAR"-URANUS 3,005, 000,000 k:m 2,'135, 000, 000 k.m0

.Kep!eys seconcfJa11lTycho's data on the orbit of Mars allowed Keplerto figure Mars' speed, regardless of where Mars was inits orbit. Looking over the speeds of Marsat different parts of the orbit, Keplerfound something quite unexpected: TheSltJCE A PLANG TRAVelS FASTE12speed of Mars is not always the same!NEAR. P(R.II-\WOt0, IT TRAVRS AWhat's more, Kepler noted that the speedchanges in a regular pattern. When MarsGREATER DISTANCE IIJ O!JE MOI\lTHpasses perihelion, it is moving fasterTHAIJ IT DOE'S l\leAR AR-iELION Itothan at any other point in its orbit.O!JE MOf\)TH. 0Then the speed decreases until it isoslowest at the aphelion. Mars picksup speed until it returns to perihelionagain.Kepler found the same behavior tobe true of all the planets he studied.A planet's greatest speed occurs atperihelion and its least speed ataphelion.1MOrJTH•1 MOt\lTHKepler decided to draw an imaginaryline from Mars to the Sun. As the planetmoves, the line sweeps out an area ofspace. Kepler discovered that Marsmoves so that it always sweeps out thesame amount of area every day. Hefound that other planets also sweepout areas in a similar way: that is,each planet moves along its orbit sothat an imaginary line joining theplanet and the Sun sweeps out equalareas in equal times. This is Kepler'ssecond law.MO~T~;j!&t~w?,,·o".).r~J(:~{E:~.]~JEij[0Fto:MomHa . '1 0o 0o TI-\E. AREAS OF e,OTHSHADED REGIONS ARt;: TI-\1OSAMe.(Exercises:1.a. Use cardboard, thumbtacks and stringto draw several ellipses.b. Label one focus the Sun.c. Choose a portion of the ellipse torepresent the distance traveled bya planet in one unit of time. Drawline segments to join the endpointsof the chosen arc to the Sun. Shadein the area.C .SUIJ• ~-----(108IDEA FROM:The Universe in Motion, Book 2, The University of Illinois Astronomy Program

.KepIeY3Jaw( COt..lTlt-)UE,D )secondd. Mark another point P on the ellipse.e. How far would the planet travel(counterclockwise) from P in 1unit of time? Mark its new locationon the ellipse.Hint:Students should draw a line segment from the Sun to P.Using Kepler's second law,this line should sweep outan area equal to the shadedportion when the planetreaches its new location.Use counters or transparentgrid paper. Measure theshaded area, or trace itand cut up the drawing.Construct the same area from P.. ::;:----- -~ -c.:..:--c::::-- - -_- ---; p2. The Jones family bakes pizza in anelliptical pan. Mom slices the pizzafrom one focus. She has just cutRon's slice and made the first cutfor Bill's piece. Where should shemake the second cut if both boys areto get the same amount of pizza?FOCUSIDEA FROM:The Universe in Motion, Book 2, The University of Illinois Astronomy Program109

(Kepler was not content with his firsttwo discoveries. He looked for a rulethat would relate the average distanceof a planet from the Sun to the time theplanet takes to complete one orbit.From Tycho Brahe's observations,Kepler was able to find the time for acomplete orbit (orbital period) of eachplanet. From a scale model (It's A GreatSystem) he was able to determine theaverage distances of the planets from theSun in astronomical units. The table tothe right shows the orbital periods andaverage distances of the planets that wereknown in the 1600's.Suppose a planet has an average distance fromthe Sun of 4 a.u. Use the graph to make a guessabout its orbital period.It took Kepler seventeen years before he discoveredthe relationship between the orbitalperiod of a planet and its distance from the Sun.Cube the average distance (4 a.u.) of the imaginaryplanet above. Now square its orbital period.The two numbers are the same!Pick another point on the curve. Suppose aplanet has an average distance of 5 a.u. Thegraph suggests that its period is between 11 and12 years. Let's try Kepler's discovery on thesenumbers:Cube the average distance: 5 x 5 x 5 = 125PLAt-JE.T PERIOD (YEARS) DISTANCE (A.U.)ME.RCURY 0.2."+\ 0.38'1VENUS 0.40\5 0.723EARTH 1.00 1.00MARS 1.88 \.52JUPITE.R 11.9 5.2.0SATURtJ 2.9.5 9.55Do you see any patterns in the table? Is there a relationship between the distancesof the planets from the Sun and their orbital periods?The facts in the table were used to plot apoint on the graph for each planet.Notice the regularity about them. They seemto lie on a smooth curve. It appears that if aplanet has an orbit of a certain size, then itsperiod will be a certain definite number of years.Square 11: 11 x 11 = 121A number a little larger thanwith the graph?Square 12:11 will give&),'>;) 131--1--1--I--HH-I­>- l~ I-+-+~I-+--t.l-+-!­:2- "I-+-+I-I-+-+-+-!­2::> ,. I-+-+I-I-+-H--+-!­(jJ91-+--+--+-I+-+-+--+-~C>281-+--+--+-I1-+-+--+-­::>~7f--j-+-+-fI;-t--j-+-~.I-+-+-t;,,+-+---j-1-­8 51--+-+--'1-++++_t:!'+I--+--f-14-+-+-+-I1­0. 31--+-+-++-+-+-11­...J 21--+-f-l-++-+-+-I1-~~\o "'--'-~"-'3'-,>t--:5~..I.G~~'--:'812 x 12 = 144AVERAGE DISTAJJCE F'ROM5UI\J llll A.U.us 125 when squared.Does this agree(Kepler's third discovery--the square of the period in years equals the cube of theaverage distance in a.u.--seems to work! Unlike "Bode's Law," Kepler's discovery does (work in all cases; Isaac Newton proved this.110IDEA FROM:The Universe in Motion, Book 2, The University of Illinois Astronomy Prograrr.

Exercises:1. An object is in an orbit at an average distance of 3 a.u. from the Sun. Read itsapproximate period from the graph. Next try cubing the distance and squaring theperiod. Do your results roughly agree?2. A spacecraft orbiting around the Sun has an orbital period of three years. Find itsaverage distance from the Sun in two different ways. Hint: 1) Kepler's law and2) the graph..•...-.. . ... _. .. ..•..... ..IMAGINARY PLAtvET ORBITAL PEelOD (YEARS) AVERr>.G£ DISTANCE FROM SllIJ (AU)3. Fill in the chart using Kepler's discovery. Check your results with the graph.V 2xyz1054. A far-out object moves around the Sun. The straight-line distance from the perhe1ionpoint of its orbit to the aphelion is 200 a.u. How long does it take for the objectto go around the Sun once?IDEA FROM:The Universe in Motion, Book 2, The University of Illinois Astronomy Program111

How fast is each planet traveling in its orbit? Doesa distant planet move more quickly or more slowly alongits orbit than a planet closer to the Sun?You already know that a planet travels slower when itis at aphelion than when it is at perihelion. As shownir the diagram it moves along its orbit so that an imag~inary line joining the planet and the Sun sweeps out2qua1 areas in equal times. The diagram is exaggeratedsince the orbits of the planets are nearly circular.Pretend that each planet does move on acircular orbit that is centered on the Sun.Because of the law of equal areas, theplanet would move at a constant speed.By assuming that the orbits of theplanets are circular, we can use Kepler'sthird law to find the average speed ofeach planet in orbit.OJ-l£MDJ-lTHEQUAL TIME::'>,EQUAL AREASo = o 0SIr-JCE. THE. QISTAN.CE. I~G£EATER I-\t:.RE.- FOR 010E.MOmH, THE. SPCED OFTHE PLANET I" GREATE-I2.III A CIRCLE THE EQUALARE-AS MAKE THI;:. ARC.'::>(DISTAl.\CE) EQUAL, SO THESPEE.D IS CONSTANt.(Remember:SPEED =DISTANCE TRAVELEDTIME TAKEklI. Find the average orbital speed of the Earth.(a.b.The average distance of the Earth fromthe Sun is 1 a.u. or 149,600,000 km.If we assume that the Earth's orbit isa circle, 149,600,000 km is the radiusof the circle. What is the circumferenceof the orbit? His about 940,000,000 km.c 0Ro°0C~ 2.1TR.CIRCUMFEREI.lCE ;RADIUS TIMES 2. 1T1T ~ 3,14-II.112c. The orbital period of the Earth is 1 year.Sospeed = 940,000,000 km1 yeard.IDEA FROM:1 yearspeed =365 days = 8760 hours = 525,600 minutes=,9"-;4:-;0~'.o:.O~00~,~0",O.o:.O-,k""m,,,;-= =,9"-;4:'.;0~'.o:.O",-00"--"k,,,,m,-;- =31,536,000 seconds 31,536 secondsC= 2.-rr >

c. The average speed of Alpha is 8 x n a.u. 1 x n a.u.8 years 1 yearaverage speed of the Earth, =2~x~n~a~.~u~.~= 30 km we find1 year 1 second'orbital speed is 15 km 11 second' on yhalf that of the Earth's.By comparing this to thethat Alpha's averageIII.Do the outer or inner planetsfollowing exercises will help1. Study the table to theright. Compare the firstand third columns. Howis a planet's distance tothe Sun related to itsorbital speed?2. Compare the first andsecond columns. Can youfind a relationship betweenthe numbers in thetwo columns?move faster than the Earth along their orbits? Theyou find the answer.123PLAtJHDISTAtJCE FI2ACTIOtJ OF EARTHS AVERAGE ORBITALFROM 5UtJ ORBITAL SPEED SPE£D (km Ieee.)GAMMA 2- GOEARTH 1 0.,\,1,. I 30ALPHA 4- o..u... ..L \5BETADELTA9 Q,.U,.Ie O,.U,.'2.I'3\03. What is the average orbital speed of a planet 16 a.u. from the Sun?4. How far from the Sun is a planet if its average speed is twice that of the Earth?5. What is the orbital period of a planet whose speed is half that of the Earth's?Hint: Find the planet's distance from the Sun and use Kepler's Third Law.6. What would you predictabout the orbital speedof an object around theSun if it is infinitelyfar from the Sun?7. How fast do the otherplanets in our solarsystem travel along theirorbital paths? Use thecalculators and use therelationship in #2 tohelp you complete thetable.PLI\l0E.TMERClIIl.YVEt0USDISTANCE FRACTIO~ OF EAR.THS AVERAGE ORBITALFROM ~Ut-J ORBITAL SPEED SPEED (km /e.ec.).39o,.u..7'2- o..u,.EARTH \.000..U,. 30MARSJUPITERSATUR.NURAP0US1.52.0..u..5.2.0 Q,.U,.9.54o..u,.19.\6

COMETS ANDCONIC5 ECTIQt\l5(From Kepler's discovery (see "Hep" KepZer Didn't Go Around In CircZes) we knowthat each planet's orbit is an ellipse. Ellipses belong to a special group of curvesknown as the conic sections. These curves have interested mathematicians and astronomersfor years.A conic section is the curve of intersectionbetween a hollow cone and a plane that cuts throughit. The pictures show some conic sections.i,, /,,--d~\lPERBOLA, '" , ", ,,//', , ,,,,, ,,PARABOLAWhy do the planets have e111ptica1 orbits? Newtondiscovered that ::he gravitational pull of the sun onthe planets causes their orbits to be elliptical. Hethen wondered whether comets were affected by thegravitational pull of the sun·and if so he predictedthat their orbits would be conic sections. In particularif the comets had nearly circular orbits thevwould be visi~le at regular and somewhat frequentintervals. Edmund Ha1iey supported Newton's beliefby discovering; that the comet he was observing wasthe same one that was seen 75 years earlier.Halley's comet· last appeared:f,n 1910 and is dueagain about the spring of 1986.---Cb-~'-,\/---HYPE.RBOLA~ ~~GO1980 1970TIll: ORBIT Of" WALU:VS COMET(We now know that the orbit of a comet will be anellipse, a parabola or one branch of a hyperbola asshown in the bottom figure. It is believed that mostcomets travel on elliptical orbits but that the orbitsare so stretched out that the portion we can seeappears like a parabola.ELLIPSE• SU~PARABOLAI-lYPtRlOOLAThe following two pages describe methods that canbe used to draw ellipses, parabolas and hyperbolas. (114

1) A Method for Drawing EllipsesCOMETS ANDCONICSECTIONS(PAGE: '2-)a) Make or obtain apage of concentriccircles (circleshaving the samecenter).b) Draw a large circle on a piece ofpaper and pick a point P insidethe circle (not the center).c) Using the page of concentriccircles, arrange a circle sothat it passes through P andis tangent to (just touches)the large circle you drew.See the dotted circle in thefigure to the right. Markthe center of the dottedcircle." ...... ~-~ ............ ,, . ,, '\',P ,,/..... _--_ .... '", "'Cd) Repeat (c) with several differentsizes of circles. Connect thecenters to form a smooth curve asshown in the figure to the left.The curve is an ellipse.115

COMETS ANDCONICSECtIONS(PAGE. 3)2) A Method for Drawing Parabolasa) Draw a line t b) Using the page ofat the bottomof the sheetof paper.concentric circles,(see 1 (a) arrange acircle so it isChoose a point p tangent to line -e.P about 4 cmand passes throughQabove the line. point P. Mark itscenter on the paper.c) Repeat (b) with several differentsizes of circles. Connect thecenters with a smooth curve. Thecurve is a parabola.3) A Method for Drawing Hyperbolas(a) Draw a circle with .pa radius about 2 cm.Label its center C.Pick a point P outsidecircle C.b) Using the page ofconcentric circles,mark the centers ofseveral circles whichare tangent to circleC and pass through,,point P. Connect thecenters in a smoothI,,\\P~ .. ­curve.;;--;",-..-~-;::; ...." ''I" 'I , ,I , ', ,,"G, , ,c)poTrace circle Cand point P onanother sheet ofpaper.centers of circleswhich pass throughP, are tangent tocircle C and havecircle C "inside"them.Connect thecenters in a smooth curve~d) Put the two, curves together,,, ,, on the same.Csheet of paperas shown.Thisis a hyperbola.p... _--(116

Our ancestors spent much time studying the motionsof the sun, moon and stars. They divided the stars ofthe night sky into groups called constellations. Onegroup of constellations, the zodiac, received specialattention.In figure 1 the solar system is placed at the centerof the celestial sphere (an imaginary large hollowsphere which appears to have the stars placed on itssurface). The twelve constellations of the zodiac liein a ring around the solar system. ("Zodiac" means"ring of animals.") The constellations of the Zodiacare listed to the right with their symbols.Although we know that the earth revolves around thesun, from earth the sun appears to move against thebackground of the stars. To help you understand theapparent motion of the sun, have a friend who representsthe sun stand in the middle of a room. As you walkslowly around the person, observe how the person appearsto move around against the background of the room.Similarly the sun appears to wander among the con-stellations of the zodiac. The apparentamong the stars is called the ecliptic.ferent times of the year the sun will appear to be indifferent constellations of the zodiac.THEZODIACFIGURE 1Constellations of the ZodiacAries (the ram)crTaurus (the bull) ~Gemini (the twins) 11Cancer (the crab) 69Leo (the lion)Virgo (the virgin)Libra (the scales)cf\.W:::C::l1\.,Scorpius (the scorpion)Sagittarius (the archer);£!Capricornus (the goat) VfAquarius (the waterI!WI/V\IINbearer)Pisces (the fish)*@LEOTAURUSA ~.GEMlt.J1C.A.OCER.0SUI\)•EARTHARIES'\,FIGURE: 2.E.CLIPTIC~CORPIUS• •path of the sunThus at dif--"'-SA(:.ITTARIU$CAPR.ICOR~U'?>GRAPH Ie FROM:Modern Space Science, by F. E. Trinklein and C. M. Huffer,Copyright© 1961, Holt, Rinehart and Winston, Publishers.Permission to use granted by Holt, Rinehart and Winston, Publishers 117

Study figure 3.Tl--4EZOD~A~(CmlTII0VE.O)In what constellationsdoes the sun appear between April and July?(April~1"MY0kJune118born.MayThe chart below can beused to find your sign ofzodiac.JulyAstrologers are interestedin each person's sign of thezodiac. Your zodiac sign is GE.MIW\ «fthe constellation in which the '}Isun appeared* when you wereMark your birth date onthe inner circle.Estimatethe correct point for theday of the month.Draw a ray from the markthrough the sun to the outercircle.The ray points toyour sign of the zodiac.*Due to the spin of theearth on its axis, theposition of the sun inthe constellations ofthe zodiac for anyparticular month isslowly shifting. Thepositions that astrologersuse in determininga person's sign of thezodiac are no longeraccurate. Sincefigure 4 gives a person'sastrologicalsign, it does notgive the actual positionof the sun in ourskys. Check a currentstar chart to find thesun's actual positionin the zodiac.the ~CJ\-l\lCER.69TAURUS~LE.OIJOV.ARIESVIRGO'YOCT.PISCESSEPT.0SU lVLIBRA-----l-X-Q;PRILMAR.H~FIGURE: '3JULYqUARlUSSCORPIOUG,MAYJUNECAPRICORWU5~f,SN;ITTARIUS"11VneURE 4(

The polar chart to the right can beused to plot the north circumpolar stars(stars near the north celestial pole thatare visible throughout the year). Thischart describes the view of the celestialsphere that you would see by standing atthe north pole and looking directly up tothe north celestial pole. The hour circlesappear as straight lines coming fromthe pole. The parallels of latitude appearas concentric circles (circles eachhaving the same center).Gn~~--~~~~~.1-+-+-1-+--11811.1) Each star of the Big Dipper in the constellation Ursa Major has been placed onthe star chart in its proper position according to its right ascension and declination.The location of S (delta) is about 12 hours 15 minutes right ascension5S0N declination. 0o= = oaS is about t of the way fromthe 12 hour to the 13 hour circle.Write the locations of the other stars in the Big Dipper.right ascension declination0

1TI S1r)l~ti~~~~~v~~( COIJTIIJUED)(\2) Plot the stars of Ursa Minor and Cassiopea on the star chart.Ursa MinorCassiopearight ascensiondeclination0

C£OJSA2Yalternate interior angles. When twolines are crossed by a third line,many angles are formed. Angles 1 and4 are alternate interior angles. Soare angles 2 and 3.1 :

ASTRONOMY GLOSSARY cir-mercircumpolar star. A star near acelestial pole which is visible fromone position on earth throughout theyear. The star is always above thehorizon as viewed from that position.concentric circles. Circles in a planeall having the same center.conic seotion. The curve of intersectionof a hollow cone and a plane.Circles, ellipses, parabolas, hyperbolasare conic sections.oonstellation. A group of stars towhich a name has been assigned.A circle on the celestialpasses through both ce1esoresoentmoon. A phase of the moon inwhich the lighted portion is less thanhalf of a full moon.deolination. The angular distance,measured in degrees, of an objectnorth or south of the celestialequator.diameter. A line segment passingthrough the center of a circle whoseendpoints are on the circle.ecleptio. The apparent path of thesun among the stars.ellipse. The curve of intersectionformed when a plane, not parallel tothe base, cuts completely through ahollow cone.anellipseequation of time. The difference beasundial and timetween time read onread on a clock.equatorial system. A method for locatingobjects on the celestial sphere.equinox. One of the intersections ofthe ecliptic and the celestial equator.gibbous moon. A phase of the moon inwhich the lighted portion is more thanhalf but not completely all of a fullmoon.gnomon. The upright portion of a sundialthat casts a shadow.hemisphere.hour oircle.sphere thattia1 poles.Half of a sphere.hyperbola. The curves of intersectionformed when a plane cuts through bothnappes of a hollow cone.hyperbolalatitude. A number in degrees thatgives a point's angular distance northor south from the equator. Latitudeis used in giving locations on theearth.light year. The distance light travelsin one year; about 9.5 x 10 12 kilometres.longitude. A number in degrees thatgives a point's angular distance eastor west from the prime meridian (ameridian passing through Greenwich,England). Longitude is used in givinglocations on the earth.magnetic deolination.ference between trueic north.The angular difnorthand magnet-meridian (celestial). A circle in thesky that passes directly overhead andthrough the north and south celestialpoles.122

ASTRONOMYGLOSSARYmer-surmeridian (on the earth). A circle onthe surface of the earth that passesthrough the north and south poles ofthe earth.orbit. The path of a body that is revolvingaround another body or point.paraboZa. The curve of intersectionformed when a plane cuts a hollow coneparallel to a line in the cone.aparabola --*+1paraUax angZe. An angle formed at astar by two rays, one going toward theearth, the other going toward the sun.sunearthparallaxangleparaUax shift. The apparent angularshift of an object on the celestialsphere when viewed from two differ~ent points.paraZZeZ Zines. Lines in a plane thatdo not intersect.paraZZeZs of Zatitude. Imaginarycircles on the earth's surface thatare "parallel" to the equator. Theyare used in telling any point's angulardistance, in degrees, from theequator.parallels of latitude4- equatorparalZels of decZination. Imaginarycircles on the celestial sphere thatare parallel to the celestial equator.They are used in telling any point'sangular distance in degrees from thecelestial equator.perihelion. The place in a planet'sorbit when it is closest to the sun.perigee. The place in an earth satellite'sorbit when it is closest tothe earth.period. A time interval betweenrepetitions; the time required forone complete revolution.perpendicuZar. Two lines are perpendicularif they meet at right angles.phases of the moon. The lighted portionof the moon seen from earth; fullmoon, half moon, crescent moon, etc.prime hour circZe. The hour circle onthe celestial sphere that passesthrough the vernal equinox.range finder. An instrument which iscalibrated to give the distance todistant objects.right ascension. The angular distanceof an object measured eastward fromthe prime hour circle.sidereal day.the earth towith respectThe time required formake a complete rotationto the stars.similar triangles. Triangles that havethe same shape, but not necessarilythe same size.solar day. The time required for theearth to make a complete rotation withrespect to the sun.surveyor. A person who measures distancesand angles on the earth toestablish boundaries.123

ASTRONOMYGLOSSARYtan~zodtangent. Two circles in the sameplane are tangent if they touch inexactly one point.transit. An instrument used by surveyorsto measure the angular separationbetween two distant objects.zenith. The point on the celestialsphere directly above the observer.zodiac. A band on the celestial sphere,centered on the ecliptic, that containstwelve constellations.(transversal.other lines.A line that crosseS twotriangulation. A method of measuringdistances indirectly that is basedupon similar triangles.vernal equinox. One of the two intersectionsof the celestial equatorand the ecliptic. The sun is at thispoint on the first day of spring.north celestial poleeclipticvernal equinox __~__~celestial equatorvertical angles. When two lines meetangles are formed. Angles I and 3 arevertical angles; so are angles 2 and 4.14 23waning. The moon is waning when thelighted portion that we see is becomingsmaller; when the moon is changingfrom a full moon to a new moon.watt.A unit of power.waxing. The moon is waxing when thelighted portion that we see is becominglarger; when the moon is changingfrom a new moon to a full moon.124

NUM££ZCAL I1ATA1. Cons tants11 ""3.14165speed of light"'" 300,000 km/s or 3 x 10km/s8astronomical unit (a.u.) ~150,OOO,OOO km or 1.5 x 1012light year ~9,500,OOO,OOO,OOOkm or 9.5 x 10kmkm1 circle contains 360 degrees (360°)1° contains 60 minutes of arc (60')l' contains 60 seconds of arc (60")1 mile ~ 1. 61 kilometresII.Data for the PlanetsAverage distance from Average distance fromPlanet Symbol the sun in a. u. the sun in km Diameter in kmMercury ~.39 57,900,000 4,878Venus~.72 108,200,000 12,112Earth $ 1.00 149,600,000 12,756Mars if 1.52 227,900,000 6,800Jupiter2+5.20 778,000,000 143,000Saturn "f29.54 1,427,000,000 121,000Uranus .. 19.18 2,870,000,000 47,000~Neptune 30,06 4,497,000,000 45,000WPlutot239.52 5,912,000,000 "'6,000III.Diameter of the sun 1,392,000 kmDiameter of the moon 3.476 kmAverage distance from the earth to the moon 384,400 kmData from Exploration of the Universe, 3rd ed.125

(((

to MATHEMATICS AND BIOLOGYBiology--a life science--included in a mathematics resource book? Biology--thestudy of plants and animals--included in a mathematics resource book? Biology--thestudy of organisms and the systems which allow the organisms to function--includedin a mathematics resource book? Yes, biologists and mathematicians are in agreementthat mathematics can be used as a powerful tool to explain many of the concepts ofbiology. Why do the inherited characteristics of plants and animals follow such apredictable pattern? Why isn't there a giant amoeba or earthworm? Why does a personget tired after many repetitions of the same task? Why are the surfaces of asponge, the lungs, the lining of the intestine, and thesurface of the brain so irregular instead of smooth?Mathematics can help to explain the answers to thesequestions.In this section, MATHEMATICS AND BIOLOGY, it isnot necessary for teachers or students to have a largeknowledge of biological concepts to be able to do theactivities. Most are simple ideas that allow studentsto investigate properties that relate to themselves andto their surroundings.The first part includes,...--------------------------------, activities which allow studentsto investigate theirreaction time, lung capacity,muscle fatigue, pulse rate,body temperature and the effectof exercise, peripheralvision, etc. Also includedis an interesting theory ofbiorhythms which attempts toexplain the ups and downs ofeveryday life in terms ofone's birth date.127

INTRODUCTION MATHEMATICS AND BIOLOGYAnother part includes elementarywork on genetics. The observation ofphysical traits in themselves, parentsand classmates allows students to developan intuitive feeling about thetraits passed down from parents to offspring.Using Mendel's experimentswith the garden pea, the basic ratiosof inherited traits are identified.Blood typing and a simulation usingcolored squares are also used to providestudents with an elementary knowledge ofthe complicated science of genetics.ATTACHEDUNATTAcHEDSome interesting and thought-prov9k-,ing activities on the concept of sizecomprise another part of this section.Students compare properties of smooth orbumpy surfaces.They also investigatethe relationships of linear, area, andvolume measurements as they apply to the (proper size of animals. A recommended ,reading for teachers (and students) is anessay by J. B. S. Haldane entitled "OnBeing the Right Size."The essay can befound in Volume 2 of The World of Mathe-matics, edited by James R.Newman or inVolume 2 of Readings in Mathematics,edited by Irving Adler.Throughout this section the intent was to provide activities that would be interesting;would not require an extensive, previous knowledge of biology; would not requireelaborate materials; would allow students to learn more about themselves; andmost of all, would show how mathematics can be useful in the study of biology.128B\oodl'"ootWhat do you notice about the number of petals in these flowers?(

MATHEMATICS AND BIOLOGYTITLEPAGETOPICMATHTYPEThe Human Body132Using measurements of thebodyPracticing computationskillsWorksheetHand 'N' Hand, Arm'N' Arm134Comparing hand sizesDet~rmining degree ofhand dominanceFinding ratiosComparing measurementsof the bodyTeacher ideaActivity cardWhat's Your Reach?136Comparing height andreachCollecting and analyzingdataTeacher directedactivityHow's Your ESP?137Exploring extrasensoryperceptionCollecting dataIntroducing probabilityActivity cardAmazing138Exploring learningabilityCollecting and analyzingdataActivity cardLung Capacity139Determining lungcapacityTaking measurementsFinding ratiosTeacher ideasYour Heart Is ABloody GoodMachine140Using facts about theheart and bloodPracticing computationskillsWorksheetYour Heartbeats141Using facts about theheartPracticing computationskills with large numbersWorksheetMy Heart Throbsfor You142Using pulse to determinephysical fitnessCollecting and analyzingdataActivity cardStep Right Up143Using pulse to determinephysical fitnessCollecting and analyzingdataInterpreting data from achartTeacher ideasMuscle Fatigue H5 Determining muscle Collecting datafatigueGraphingBody Temperature 146 Determining the effect of Collecting and analyzingexercise or smoking on databody temperatureSpy on the Eye 147 Determining the rate Collecting and analyzingof eye blinksdataTeacher ideaActivity cardActivity cardActivity card129

TITLE PAGE TOPIC MATH TYPEI See It 1Lf8 Determining angle of Measuring angles Activity cardperipheral vision and Measuring distancesdepth perceptionGaining or LosingMass149Determining calorie needsDetermining body typeDe·termining mass of bodypartsCollecting and analyzingdataInterpreting data from achartTeacher ideasWhat's Your Type?156Determining body typeInterpreting data from achartWorksheetActivity cardBody CountingSys'terns157Using symmetry of thebodyIntroducing differentnumeration systemsTeacher pageLet's Split Hairs158Investigating microscopicorganismsMaking a scale drawingInterpreting small numbersActivity cardSurface Area orVolume?159Investigating surfacearea and volumeDetermining mass is dependentupon volumeWorksheet(Sink or Swim?160Investigating surfacearea and volumeDetermining mass is dependentupon volumeWorksheetBumpy or Smooth?161Investigating surfaceareaDetermining bumpiness isdependent upon surface areaWorksheetMonster Earthworm162Determining the effecton body surface area andbody volume caused bymUltiplying body dimensionsComparing measurementsWorksheetInherited Traits163Introducing concepts ofgeneticsUsing fractions, percents,ratios, and/or probability"to analyze dataTeacher pageAnalyzing Traits165Analyzing physical traitsCollecting and analyzing dataTeacher ideasA Model forInheritance168Developing a model forthe inheritance ofphysical traits'Collecting and analyzing dataActivity cardWhat Type Are You?169Determining blood typeCollecting and analyzing dataTeacher ideaJust Like Peas ina Pod171 Using Mendel's conceptsto determine physicaltraitsCollecting and analyzing dataInterpreting da"ta from a chartTeacher idea(130

1. Your body contains 206 bones.1About 7 of these bones are inyour head.2. Muscles make up about .4 of aman's body weight.absorbed into the blood stream.3How many m of oxygen is this?4. An eye blinksabout 25 timeseach minute.About how many times does itblink in a day (24 hours)?5. Your intestines (large andsmall) are about 7.5 metreslong..8 of thislength.Yoursmall intestineis aboutHowlong is yoursmall intestine?What dothe muscles of a 82 kilogramman weigh?About how manybones are there in your head?3. In one day, a man breathes in3about 12 m of air. About.05 of this is oxygen that isSOURCE:6. A man's brain is between .027.and .03 of his body weight.Between whattwo numbersis the brainweight of a68 kilogram1The brain of a baby is about 10its body weight.About howmany grams does the brain of a4 kilogram baby weigh?8. Your heart beatsabout 80 times aminute.notation.About howmany times does itbeat in a day?9. The body of an adult containsabout 5 litres of blood.A blooddonor usually gives t litre ofblood.What fraction of hisblood does the donor give?10. Your body contains about 165,000kilometres of blood vessels.Write this number in scientificInvestigating School Mathematics, Second Edition, by Robert E. Eicholz, Phares G. D'Daffer,Charles R. Fleenor, Copyright © 1976 by Addison-Wesley Publishing Company, Inc.School Mathematics I, by Robert E. Eicholz, Phares G. D'Daffer, Charles R. Fleenor,132 Copyright © 1976 by Addison-Wesley Publishing Company, Inc.All rights reserved. Reprinted by permission.((

liNIl N"MAN•••y(COIJTllJUED)PROBLEMS-ESTIMATING BODY HEIGHTSHumerusHeight (centimetres)Male(2.894 x length of humerus) + 70.6400.271 x length of radius) + 85.925(1. 880 x length of femur) + 81. 305(2.376 x length of tibia) + 78.663FemaleAnthropologists, using a single bonefrom a human skeleton, can estimate quiteaccurately the height of a man or womanwho lived many centuries ago. Forexample, if an anthropologist finds a27.94 em radius from a caveman, he canestimate the height of the caveman asfollows:(3.271 x 27.94) + 85.925or 177.316 cm. Thus, the man was about180 cm tall.1. A 48 cm femur was found. About howtall was the man?2. The length of a humerus bone from awoman who lived 20,000 years ago is32.25 cm. What would be a goodestimate of the woman's height (tothe nearest centimetre)?3. The skeleton of a Neanderthal manwho lived about 50,000 years agocontained a tibia 38.1 cm long.Give an estimate of the man's height.(2.754 x length of humerus) + 71.4760.343 x length of radius) + 81.224(1. 945 x length of femur) + 72.845(2.352 x length of tibia) + 74.7754. A man is 181 cm tall. Using thetable, accurately estimate the lengthof the bone from the knee joint tothe ankle joint.5. Measure, as accurately as you can,the length of the radius and humerusof an adult and check the accuracyof the table.6. A student's height was found to bethe same as the distance fromfingertip to fingertip when his armswere outstretched. How much doesyour height differ from this distance?7. One student's height in inches was8.7 times the length of his middlefinger measured to the nearesttenth centimetre. How much doesyour height differ from this distance?SOURCE:Investigating School Mathematics, Second Edition, by Robert E. Eicholz, Phares G. O'Dafter,Charles R. Fleenor, Copyright© 1976 by Addison-Wesley Publishing Company, Inc.School Mathematics I, by Robert E. Eicholz, Phares G. Q'Daffer, Charles R. Fleenor,Copyright© 1976 by Addison-Wesley Publishing Company, Inc.All rights reserved. Reprinted by permission.133

,1.1) Keep your fingers together andtrace around one of your hands.2) Draw line segment 1 across theknuckles.3) Draw line segment 2 where thewrist begins.4) Draw line segment 3 from the topof the longest finger to thewrist line segment.5) Measure line segments 1 and 3 tothe nearest centimetre.F2.T6) Find the ratio of the length of SEGMo,n nf\(\f\line segment 3 to the length of 1VV\.)Vline segment 1-7) Compare your ratio to yourclassmates. Are the ratiosSE:.GMEtJTabout the same? 38) Label the thumb T, the indexfinger Fl, middle finger F2,ring finger F3 and little fingerF4.9)Is Fl longer, theshorter than F3?your classmates.same as orCompare withII.1) Place one end of a metre stick snugly against your armpit and measure the lengthyour outstretched right arm.2) Repeat measuring the left arm.Right arm cm Left arm cm3) How do the lengths compare?4) What results did your classmates get?IDEA FROM:134Modern Life Science, by F. L. Fitzpatrick and J. W. Hole.Copyright © 1970 by Holt, Rinehart and Winston, PublishersProbing the Natural World, Volume 1, and Investigating Variation, Intermediate ScienceCurriculum StudyPermission to use granted by Holt, Rinehart and Winston, Publishers, and Silver Burdett General LearningCorporation(

III.Between the ages of 2 and 4 a child usually begins to establish a preference forusing either the right or left hand for performing tasks. The following activitycan be used to establish handedness and also the degree (or dominance) of handedness.Type a page of zeros and furnish a dittoed copy for each student.1) As one student acts as a timer another student crosses off zeros on thesheet for 30 seconds. An X is the preferred method for crossing off thezeros.2) Repeat, this time using the other hand. Record both results in a table.3) Change roles and have theother student get and recordhis scores.4) Which is the dominant hand?5) Divide the smaller numberinto the larger number todetermine the degree ofhandedness (dominance).RIGHILEFTFor Example: RIGHT 50LEFT 25225150The right hand is twice asdominant as the left hand.6) Does anyone in the class have a degree of dominance that is almost I?7) What could this mean?IDEA FROM:Modern Life Science, by F. L Fitzpatrick and J. W. Hole,Copyright © 1970 by Holt, Rinehart and Winston, Publishers.Probing the Natural World, Volume 1, and Investigating Variation, Intermediate ScienceCurriculum Study.Permission to use granted by Holt, Rinehart and Winston, Publishers, and Silver Burdett General LearningCorporation135

Tape a strip of butcher paper to the wall to mark the heights and reaches of thestudents in your class. Label each mark with the student's name. Mark and label eachstudent's foot length. Measure, to the nearest centimetre, each student's data andrecord in a table........~E(GHTREACHFOOTLDJGTI-\If the table is drawn on the board or on a poster, have students examine the datafor patterns. The data should show that the heights and reaches are approximately thesame for each student. In the two extra columns have each student compute, to thenearest hundredth, the ratio of height to reach and the ratio of height to foot length.The former should be about 1.00 and the latter will probably show no pattern. Takingthe average of all the ratios of height to reach will probably be very close to 1.00.The data in the table can beused to make a bar graph. Provideeach student with a small squareand let him glue the square in theappropriate column on a graph likethe one shown here. The graphcould be used to develop the conceptsof range, mean, median andmode.:, : , : , : :I , , I : , !: I , , , I : ,I , , ,::, :I ,I ,I , '8', ,:: :, ,' I ', :B: : :a: : : : ! ' ,, ,01 : ,I: ' I, , , ,' , I , , :~ ~ ~~ iii ~ ~ ~ ~ ~ 2 f:! ~ gI\136

HOW'S YOUeFor this activity you need five cards of each of the four suits from a regular deckof cards. Shuffle the twenty cards and place them face down in a pile. Without lookingat the top card guess what suit it is, and record your guess on a tally sheet.Don't look at the card.Place it facedown to start another pile.Continueguessing and placing the cards on topof the previous card.When all twentycards have been guessed check yourguesses by turning over each card.Remember:The first card you turnover will be the 20th card you guessed.Shuffle the cards and repeat theactivity four more times until youhave guessed 100 cards.Write this ratio.CO~I2.ECT GUE'::>SES: TOTALGUESSESChances are you guessed correctly about25 out of 100. If you guessed correctlymore than 35 times you might havesome ESP.Used by permission of Allyn and Bacon, Inc.ADAPTED FROM: Doing Their Thing In Action. BiologV Series by Stanley L. Weinberg and Herbert J. StoltzCopyrlght© 1974 by Allyn and Bacon. Inc.'

This activity will see how quickly you can learn the correct path through a maze.Place the hole over the start position and count each dot you see until you get to theend. Do six trials and record the results for each trial. Repeat the activity tomorrowto see if you can recall what you learned.STARTENDl----t.((138IDEA FROM:Modern Life Science, by F. L. Fitzpatrick and J. W. Hole,Copyright © 1970 by Holt, Rinehart and Winston, Publishers.Permission to use granted by Holt, Rinehart and Winston, Publishers

Two activities are outlined below to determine a person's lung capacity.The datacollected from either can be plotted to find the distribution, the range and the averageof the volumes.The data could also be used to look at the relationship between weightand volume, height and volume, age and volume, athlete-nonathlete and volume, smokernonsmokerand volume,etCaStandard lung capacity for boys and girls is found using the following relationship:standard capacity in millilitres = height in centimetres x G23 ) or (20 \boys girls)This is just an average aud young boys and girls, especially in the growing years,should not be too concerned if they do not meet the standard.A fairly good indicatorof physical fitness is the ratio of maximum lung capacity to standard lung capacity.In general, the higher the ratio is, the better the physical fitness is.Maximum lung capacity:IUsing one deep breath blow up aballoon as much as possible.Forcethe balloon into a container filledto the top with water.Catch theamount of overflow in another container.Measure this overflow amountin millilitres to find maximum lungcapacity.IIBreathe deeply several times and thenexhale into the jar.Your air willforce water out the other tube.Measure this overflow amount in millimetresto find maximum lung capacity.ADAPTED FROM:Keeping Alive in Action Biology Series by Stanley L. Weinberg and Herbert J. Stoltze,Copyright© 1974 by Allyn and Bacon, Inc.Used by permission of Allyn and Bacon, Inc.139

.''Y07.(f2 IfEfiJ2r15.Jt. ELDOD;GOOD JM!CJff(lf£I. The human heart pumps about 60 millilitres of blood for each beat.1) If an average of 80 heartbeats per minute is used, how many millilitres of blooddoes the heart pump per minute?2) How many litres of blood is this?3) Get two large containers and fill one with water. Use a 50 ml measuring cupto see how much water you can scoop into the empty container in one minute.Measure this amount and compare it to the amount the heart can pump in oneminute.4) Try a garden hose or a faucet. Can either put out as much in one minute as yourheart?II.It is estimated that a human has about 25,000,000,000,000 (25 trillion) red bloodcells. The average diameter of a red blood cell is about .008 millimetres.1) If placed end to end, how long of a line would these blood cells make?(2) If placed in the shape of a square, how long would each edge of the square be?3) If placed in the shape of a cube, how long would each edge of the cube be?ADAPTED FROM: Keeping Alive in Action Biology Series by Stanley L. Weinberg and Herbert J. 5toltze,Copyright © 1974 by Allyn and Bacon, Inc.Permission to use granted by Allyn and Bacon, Inc.(.Well-Being, Probing the Natural World, Level Ill, Intermediate Science Curriculum Study140 Permission to use granted by Silver Burdett General Learning Corporation

By yourself or with a friendbeats in a minute.count the number of times your heart,m""l.1 year?million16 years?~ millionYou'll reach 1billion heartbeatsin ~ years.~ billion ---heartbeats in 100years.141

~ ~My HEARTTHRO S~OR Yo~Materials Needed:2 studentsStopwatch or clock with a second handActivity: 1. On your paper draw a chart like the one below.Ino..e.tlve Adive. t(e.cQve'r"'{No-me Pulse Pulse PulseSel+' P oy +ne.or Sel+' Po.rtner- Set+ R:..r+nev-2. a) Guess how many times your heart beats inone minute. bpm (beats per minute)b) Have your partner take your pulse andrecord it in the inactive column as___ bpm.c) Take and record your partner's pulse.(3. a) Run in place for two minutes.b) Record your pulse rate in the activecolumn.c) Have your partner run in place for twominutes.d) Record your partner's pulse rate.REST 5 MINUTES4. a) Record both of your pulse rates in the recovery column.b) Have your pulses returned to normal?c) Is your recovery rate faster than your partner's?( ..142

Numerous ads to eat wisely and exercise regularly encourage students to thinkabout their physical condition which, in turn, affects the pulse and recovery timefollowing exercise. In general, conditioned persons have a slower resting pulse anda slower pulse during exercise. Their pulse will recover to the resting rate quickerfollowing strenuous exercise than persons who are in poor condition. Because ofheredity some persons inherit efficient hearts with slower rates, while others areborn with relatively inefficient hearts. However, both types can be improved.Since the physical condition of an individual affects his heartbeat, pulsetests can be used to measure physical fitness. Four pulse tests are describedbelow, and tables to interpret the results are provided. Better results can beobtained from the first two tests if they are done at home with parental help.1. Pulse Lying:The pulse lying is the slowest, resting pulse of a person. The student canfind this rate by taking her pulse for 30 seconds before she gets out of bed inthe morning. If done in class, have the student lie down and attempt to completelyrelax for ten minutes. In the lying position count her heartbeats for 30 seconds.The student should continue to rest in the lying posltion for 2 more minutes andrepeat the count. If it is the same, double the count to get the pulse lying, andrecord the number. If less the student should rest longer and repeat the count.II. Pulse Standing:To obtain the slowest, resting, standing pulse have the student rise slowlyafter finishing the pulse lying test and remain standing for two minutes. Countthe heartbeats for 30 seconds and double the number to get the pulse standing.Have the student subtract the pulse lying from the pulse standing. Thisnumber is the pulse difference. By checking Table A the student can find her physicalfitness rating.APhySiCQITABLE A~~~~~p/f~':,.'U7 58 60 83 66 G9 71 73 75 77 78 79 80 82 84 8G 105Pu.l Se. Sta.ndir"IQ 4G 63 G7 G8 70 74- 77 80 83 85 87 90 9\ 92 94 96 98 101 IZ3Pu.lse Ditf'el"Elnce G 9 to 10 10 II II \I t2- 12- 12- 13 13 13 14 14 14- 15 18IDEA FROM:Phvsical Fitness Workbook, by Thomas Cureton,Copyright © 1944 by Stipes Publishing Company.Permission to use granted by Stipes Publishing Company143

144III.IDEA FROM:(COWl"INU£O)Simplified Pulse Ratio Test:(a) While sitting, have the student count and record her heartbeats forone minute.(b) Have the student face a chair (approximately 45 cm high) and step upwith the left foot, up with the right, down with the left, and downwith the right. The student should do 30 of these steps in one minute.In order to set the cadence the teacher or another student can callout "up, up, down, down" at the required speed or playa taped recordingof the cadence.(c) Immediately after completing the 30 steps, the student should sit,count and record her heartbeats for two minutes.(d) Have the student write her pulse ratio. Pulse Ratio = Heartbeats for2 minutes following the exercise : Heartbeats for 1 minute beforeexercise. Simplify theratio by dividing the firstnumber by the second, correctto one decimal place. TABLE BCheck Table B to find thephysical fitness rating.Pulse.Physico.lRo..-l:io Fi+ne.!>£, Ra.ti ngIV. Three Minute Step Test:This test is administered likethe previous test, except thestudent steps for three minutes,and the cadence is 24 steps perminute. In addition wait oneminute after the student completesthe exercise and countthe heartbeats for only 30seconds. The efficiency scoreis the ratio ofnumber of seconds stepping x 100pulse for 30 seconds x 5.6Divide and check Table C to findthe physical fitness rating.*This table is accurate for juniorhigh girls. The efficiency scoresmay ne~d to be raised for juniorhigh boys. At the grade schoollevel there is not much differencebetween boys and girls.BelolN 2.'3 Above Aver9file2.'3 - 2..7Above 2.'1TABLE C*?hysicO-\Fi+ness'Ra..+lno72 -100 Excellent J[.t-f i Ciet"1c~Sc(")\'"eAvera.~eBelow AverQ!2,e62-7\ Very Good51-f,1 GoodAI-50 I="Q.. i r31-40 Poor0-30 Vey-l.j 'PoorPhysical Fitness Workbook, by Thomas Cureton.Copyright © 1944 by Stipes Publishing Company.Permission to use granted by Stipes Publishing Company(((

,,/The information gained from this activity involving muscle fatigue can be usedfor graphing, finding mean and range, and finding percent of decrease and increase.Three students are needed per group; o~e as the "muscle" person, one as thetimer, and one as the counter.The "muscle" person places the right forearm flat on a table so the back ofthe fingertips are flat on the tabletop. He/she closes and opens the right handas fast as possible until the timer says stop, being sure the fingertips touch thepalm when closed and the fingertips touch the table when open.The timer times each trial for 30 seconds and records the tally on the chart.Between the 3rd and 4th trials the "muscle" person is given a 3D-second rest period.The counter counts the number of times the fingertips touch the table, relaysthe count to the timer, and begins the count over at 1 for each trial.After the activity has been completed for the right hand, repeat the steps forthe left hand.A chart with sample data and a graph of that data are shown below.8030 SECOtJD RIGHT UTTPERIODS HAtJD HAND'70GORIGHTLEFT~1\ ,'1 72-

I. Does a person's body temperature remain constant all day long?(a) For avals.p.m. ,period of two days take and record your temperature at four-hour inter­This can be done at 7:00 a.m., 11:00 a.m., 3:00 p.m., 7:00 p.m., 11:00and 3:00 a.m. or any other set of times that is convenient.b) For the 7:00 a.m. reading take your temperature before you get up and startbeing active.c) Record the temperatures in a table.d) What results do you see?7:00 11:00A.M· A.M.3:00 7:00 11:00P.M. P.M. P.M.3:00A.M.DAY 1DAY 2II.What is the effect of exercise on body temperature?a) Take and record your temperature.b) Exercise for 5 minutes.o(c) Take and record your temperature.od) Rest for 5 minutes.e) Take and record your temperature.oIII.Does smoking affect fingertip temperature?a) Have several smokers and nonsmokershold a thermometer in their fingertips.Record the temperature.SMOKE.RSGNON SMOKERSb) Is there a difference between thefingertip temperatures of smokersand nonsmokers?38985105,l)AVERAGE:AVERAGE: (146IDEA FROM:Physiological Adaptation, Inquiry into Biological Science, and WeI/-Being, Probing theNatural World, Level III, Intermediate Science Curricuium StudyPermission to use granted by D. C. Health and Company, American Book Company, and Silver BurdettGeneral Learning Corporation

Spy ON THE EYEMaterials Needed:Clock with a second hand2 secretive, spying studentsActivity: (1) For 1 minute count the number ofblinks your partner makes and recordblinks per minute in the table. Askyour partner to blink in a naturalway.(2) Have your partner find your rate ofblinking.(3) Move to a bright area--near thewindow, near a lamp, in the sunshine--andfind out if the blinkingrates increase.(4) Secretly find the blinking rate of4 other students, 2 girls and 2 boys.Record.(5) Do the same for your teacher.(6) Find and record the blinking rateof someone wearing contact lenses.(7) Is there any difference in the blinking rate of boys and girls?(8) Is the blinking rate of the student wearing the contact lensesfaster than the other rates? Why?(9) Use your blinking rate to find the number of blinks you will makein a day? a year?(10) If your eyelids move 2 cm in a blink (1 cm in closing and 1 cm inopening), how far will your eyelids move in a day?147

I. To measure your angle of peripheral vision:1) Tape one end of a I-metre piece of string to the floor.2) Use a piece of chalk and the string and draw part of a circle.3) Stand on the tape and have one partner stand at the other end of the stringholding card 1. Another partner, holding card 2, moves around the circle.(4) You must face person 1, stare at card 1 and tell the second partner to stop whenyou no longer can see card 2.5) Measure your angle of peripheral vision with a protractor.II. To test depth perception:1) Arrange two dowel rods as shownso they can be seen against thebackground card.2) Have a partner move rod B untilit appears even with rod A.1-T I TE.~T 2 AVE'i?AGE.RIGHTRODS SHOULD e,e: SEENLIKE THIS.oIDEA FROM:DLEFTInvestigating Variation, Intermediate Science Curriculum StudyPermission to use granted by Silver Burdett General Learning Corporation(148

GAININ.OQLOSING ()J{f:f!f3Students have surely been exposed to the term calorie. Since many students areexperiencing a growth period in grades 5-8, the study of calories and diet seems veryappropriate. Scientifically, a calorie is a unit used to measure the amount of heatneeded to raise 1 gram of water 1 Celsius degree. The relationship is described bycalories = grams of water x change in temperature (Celsius degrees). Most science bookscan provide exercises involving the above relationship. Probing the Natural World,volume 2, gives an experiment that students can do to approximate the number of caloriesin five mini-marshmallows.The unit of measure used when dealing with the calories in food is called a largeCalorie. It is equivalent to 1000 calories (kilocalorie) or the amount of heat neededto raise the temperature of 1000 grams (1 kilogram) of water 1 Co. These kilocaloriesare indicated by using a capital C (Calorie). The following chart shows a sample of theaverage daily Calorie needs.AGE.CHILD GIRL BOY WOMAlJMODERATELV10-1:2. 12.-14 15-18 12-\4 15-18 ItJI\CTI\!E ACTIVE-, "MANVERY MODERATE-LY VER.'/ACTIVE II\lI\CTIVE- ACTIVE: ACTIVE:CALORIES 2000 2'2.00 2GOO 2GOO 3000 :2.000 2/+00 3000 2'1-00 3000 4500Have students keep a diary of their meals for one week. Record Calories per mealand per day to determine if they are getting too many, too few or the appropriate numberof Calories. The charts below give the Calories for various foods. For unlisted foodsstudents may have to consult a diet book. It is important for students to realize thatproper eating does not mean just counting Calories. Such things as vitamins and mineralsare also important.IDEA FROM: Modern Science Earth, Life, and Man, by S. S. Blane, A. S. Fischer, and O. Gardner,Copyright © 1971, by Holt, Rinehart and Winston, Publishers.Well· Being, Probing the Natural World, Level III, Intermediate Science Curriculum Study.Permission to use g'ranted by HOlt, Rinehart and Winston, Publishers, and Silver Burdett General LearningCorporation 149

·"ltll•• oRLOSING(PAGE. 2)(.",-Food Measure CaloriesBeveragesApple juice I cup 120Coffee, black I cup 0Cocoa 1 cup 234Cola drinks 1 glass 105Cream (heavy) I tbsp 50Milk, choc, I cup 185Milk, skim I glass 90Milk, whole 1 glass 165Milk shake 1 glass 340Orange juice I cup 105Tea I cup 0Tomato juice 1 cup 50Cakes, Pies, etc.Angel-food cake 2" wedge 110Brownies I piece 100Chocolate layercake, fudge frost. I slice 350Cookies 1 large 120Cupcake, iced 1 med, 185Doughnut, cake 1 med, 135Doughnut, jelly 1 med, 185Pie, apple 4/1 wedge 335Pic, pecan 3" wedge 570Main DishesBaked beans-pork ~ cup 240Chicken pie 4t" diam, 535Hamburger & bun 1 med, 315Hot dog & bun 1 med, 270Macaroni & cheese ~ cup 350Pizza, serving 1 med, 185'Rice, boiled 2cup 100Soup, creamed 1 cup 135Soup, navy bean 1 cup 170Spaghetti 2cup 260Stew (meat-veg,) I cup 252Breads, etc.Biscuit 1 (2") 60-85Cornbread 1 slice 100Crackers, saltine 2 med, 35French toast(no syrup) 1 slice 140Melba toast 1 slice 20Muffin 1 (2") 100-145Pancake (no syrup) 1 (4") 60Food Measure CaloriesRaisin . I slice 65Rolls, sweet l'med, 135Rye I slice 57Waffle (no syrup) I (4" sq,) 120White 1 slice 64Whole wheat 1 slice 55CerealsBran flakes I cup 117Cooked cereals 1 cup 165Corn Flakes I cup 96Grape Nuts I tbsp, 28Puffed Rice I cup 55Rice ,Krispies 1 cup 133Fruits and NutsAlmonds I cup 850Apple I med, 75Applesauce 1 cup 184Apricot I large 18Avocad6 1 med, 360Banana 1 large 119Berries 1 cup 75Cantal0.upe 2med, 60,Cranberry Sauce 2cup 225Dates, dried 3-4 dates 115Fruit cocktail ! cup 60Grapefruit 1 60Grapes sm. bunch 55Orange 2 60Peach 1 med, 35Peanuts (roasted) 1 cup 805Pear 1 med, 50Pecans 1 cup 750Plum 1 med, 35Prunes, drieq 4 large 115Raisins i cup 115Strawberries 1 cup 55Walnuts 1 cup 655Watermelon 1 slice 45Dairy FoodsButter 1 tbsp 100Cheese 1'1 cube 110Cheese, cottage 2 tbsp 30Ice cream, vanilla 2cup 145Sherhp,rl 2cup 130TJ>,£IL!:: (CO\.1TI\.lUEO)((TABLE FROM: Well-Being, Probing the Natural World, Level III, published by Silver Burdett Company,150 Permission to use granted by Silver Burdett General Learning Corporation

GAl.I•• ORLO SING(J;fj'ff3(PAGE:. 3)Food Measure CaloriesMeat, Fish, PoultryBacon 2 slices 100Beef, roast I slice 75Eggs I med. 80Fish 3 oz 135Fishsticks 3 02 170Frankfurter I med. 155Ham (lean) 202 125Hash 302 120Lamb I chop 140Liver 3 oz 195Luncheon meat 2 slices 165Pork I chop 250Sausage I link 90Steak 30z 250Tuna, canned ! cup 115CandyCandy bar, avg. I sm. 130Candy, hard I 36Caramel I 50Fudge, plain I" sq. 115Marshmallows I 25Mints or patties I 40MiscellaneousCatsup I thsp 20Gravy 2 tbsp 55Jam, syrup, honey I tbsp 60Jello 1cup 50Peanut butter 2 tbsp 190Potato chips 10 med. 115Popcorn,lightly buttered ! cup 35Sugar ItsI' 16Vinegar 1 tsp 0-Food Measure CaloriesVegetablesAsparagus 6 spears 22Carrots, cooked 1cup 20Carrot, raw I sm. to med. 25Carrot~raisin saL 3 tbsp 150Celery 2 sm. stalks 5Coleslaw 2cup 60Corn 1cup 85Corn on the cob 1 ear 84Green beans ! cup 15Green leafy veg. Qcup 20Lettuce (head) ~ med. 15Lima beans Q cup 90Peas 2cup 60Pickle, dill 1 large 15Pickle, sweet 1 22Potatoes,French fried 6 pieces 90Mashed ~. cup 65Salad 2cup 185Sweet ~ cup 85White, baked I med. 80Radish 1 1Squash 1cup 65Tomato I sm. to med. 25Salad DressmgsFrench I tbsp 60Italian 1 tbsp 85Mayonnaise I tbsp 110Russian 1 tbsp 100Roquefort I tbsp 100Thousand Island 1 tbsp 50TABLE FROM:Well-Being, Probing the Natural World, Level III, pUblished by Silver Burdett Company.Permission to use granted by Silver Burdett General Learning Corporation151

GAI.I•• oRLOSING ~(PAGE. 4)(Calories that aren't used up as heat in the body cause a gain in mass. Many differenttypes of diets try to cut down on the number of Calories eaten while trying toincrease the amount of activity to use up extra Calories. Having a mass that is aboveaverage is becoming a serious health problem for Americans. Medical authorities classifyanyone whose mass is 10 to 19% above the average as "overweight" and anyone whosemass is 20% or more above the average as obese. These authorities estimate that about50% of all Americans are "overweight" and 33% are obese. You may want your students todetermine whether they are under average, average,or over average. Certainly some studentsmay be sensitive about chis subject.Growth charts for boys and girls with ages from 10 to 17 are shown on the next page.The average is indicated by the solid line with the band between the two dashed linesindicating the average range. The bands above and below the dashed lines indicate thetall, short, heavy. and light ranges.The following chart shows the death rate for men according to their age and mass.The rates for women are slightly lower than those shown for men. For example, the indexfor a short, under 40, 27 kg 9ver average man is 1.90. This means he is almost twice aslikely to die as a short, under 40, average man. The statistics show that it is probablybetter to ba a little under average.(AGES"+0 AI\JO OVr.RMASS SHORT MEDIUM TALL SHORT MEDIUM TALL18 kg UNDER AV£RAGE 1.15 1.\5 1.2.0 \.00 1.009kq UNDER. AVERAGE 0·95 0.90 0.90 1.00 0.95 0.95Okg AVl:RAGE 1.00 \.00 1.00 '.00 \.00 1.009kg OVER AVERAGE 1.15 1.10 1.\0 \.2.0 1.2..0 1.1018k';3 OVER AVE:l;aAGE 1.35 1.2.5 1.2..5 1.35 l. '30 1.2..52.'1 kq OVER AVERN:i,r::. 1.90 1.'+5 1.'-15 I. GO 1.50 I. 4-5(152Permission to use granted by Silver Burdett General Learning Corporation

1'751'70ISI;jlIGOwQ( 155t;:;;: 150i=~11.j.5~l- ILIO:r12 ISSW::r 13012.5.AI.I•• oRLOSING CMm3(PAGE: 5) (;3-- .... ~-,,,~~, ,, /' I---,/, / V ---~~I/.........I/ I / 1/"/ I, V / '/ 1// , ,/I/ / /,'/ / / // , ,/, ,,"GIRLS1'2.010 II 12 13 14 15 IG 1'7AGE(YEARS)G2595G53I..J :~l.I452'-' 41~ 383532292GGIRLS,:, ,,-,----,/--,/, , ,--,I,VI, ,, I / ,/,,/ /,-I I ,/ I, II, , I/ /I ,I,,I I ,, , I,, /I,/ /, ,/'/ / I,,//I , ,I , ,""~~....., ,,,2310 II 1'2. 13 Iii- I 1

GAI.I•• oeLOSINGc::rJ:f§3(PAG£ G)(Different activities require different numbers of Calories; in general the morestrenuous the activity, the more calories needed.Calories needed for kilogram of body mass per hour.The following chart gives the averageStudents can chart the activitiesthey go through on a typical day and determine the number of Calories needed for thoseactivities. Some unlisted activities will need to be compared with a similar one on thechart. They can also compare the number of Calories needed to the number of Caloriesactually eaten.Calories UsedRounded Time (per kilogram of Body MassActivity (in hours) body mass per h) (in kilograms) Calories UsedBicycling (fast) 7.5Bicycling (slow) 2.4Dishwashing 1.1Dressing and undressing .7Eating .4Playing Ping-Pong 4.4Running 7.3Sitting quietly .4Sleeping .4Standing .4Studying or writing .4Swimming 7.9Tennis 6.6(Typewriting rapidly 1.1Violin playing .7Volleyball 5.5Walking 2.0Work, heavy 5.7Work, light 2.2Total Calories Used per DayTABLE FROM:Well-Being, Probing the Natural Worid, Level III, published by Silver Burdett Company.Permission to use granted by Si Iver Burdett General Learning Corporation(154

GAI.I•• oRLOSING ~(PAGE. 7)The figure shows a person that has a mass of 60 kilograms. Various parts of thebody are labeled with the mass of that part. Determine the percent of the whole bodythat each part represents. Then use these percents to find the mass of the same partsof your body.,-,, , -," " ...., ' ,, ,,/, ,,,I, ,I,I,,, , ,,,, , , ,,,, ,,,,",...... _-_ ..., ,,,/I,,, ,,,,,,,\ ,'-4.6kg, III ,III ",4.8.'---',.... / '\, , ,,'-,,, , , ,,,,,,,,,,,,II"""1 ,1 ,, , ,, ,\ \ \I , " ,, ,,,,,, ,, i , ,,,, ,( \ ( \I 1 I II : : II 1 , II 1 I II: $.(;I I:: 3.GI:I : I II I I I: kq :: kg :I : : II I I II I I IJ ,: :, :,I I : II , I II I I It J : I' ........ ....... / " ..... - ,1, ,\ \\ , ' I, I, ,, '\ //' ...... _.....,;Permission to use granted by Silver Burdett General Learning CorporatiQn155

V:hoi's YOUQ TYPE1. Weigh yourself and measure your height. ___ pounds inches___v-./"""---" (2. Change your weight to kilograms.1 pound ~ .45 kilograms.3.Change your height to centimetres. (1 inch"" 2.5 centimetres ~4. Use the chart to determine your body type.Ma.ssin kilograms~Ho;q"+ ;n COn+;motco, GROWTH CHAR.T FOR GIRLS10 Yrs. II Yrs. 12 Yrs. 13 Yrs 14 Yrs· 15 Yrs.{TQIl 143-15S 153-IG3 157- 168 IG2-170 IG2- 173 1G4-173Avera.ge 134-142 140-152 147-156 152- IGI 154-1

BODY COUNTI NG SYSTE~1SBODY COUNTING SYSTEMS OF NUMERATION ARE USED BY A LARGE NUMBEROF PAPUAN AND NEW GUINEAN GROUPS, EACH NUMBER IS INDICATED BY THECOUNTER POINTING TO J TOUCHINGJOR CALLING THE NAME OF A BODY PART,MOST SEEM TO HAVE AN ODD NUMBER AS THEIR BASE WITH THE SAME BODYPARTS ON EITHER SIDE OF THE NOSE OR FOREHEAD BEING INDICATED (INREVERSE ORDER) ON THE WAY UP AND THE WAY DOWN.TIlE EAST AND WEST KEWA SYSTEM WHICH USES BASE q7 IS GIVEN BELOW:1 little finger 472 ring finger 463 middle finger 454 index finger 445 thumb 436 heel of thumb 427 palm 418 wrist 409 forearm 3910 large arm bone 38{6a16 shoulder bone 32~' .....:2~O.:

~1AGN IFI EDARE DRAWN Te SCALE.spoRe 0;:. CCl'YYY'oNFIELD T")USHROoMGET A SCIENCE BOOK OR ENCYCLOPEDIA TO FIND THE SIZE ANDSHAPE OF OTHER THINGS. DRAW PICTURES MAGNIFYING YOUR THINGSIN PROPORTION TO THIS HAIR .HERE ARE SOME SUGGESTIONS:1) RED CORPUSCLES2) VARIOUS SMALL INSECTS (LOUSE, FLEA, ETC.)3) GRAIN OF SAND (OR ANY OTHER CRYSTALS)4) ALGAE (VARIOUS TYPES)5) FUNGI (MUSHROOMS, SMALLER FUNGI)6) DIATOMS7) CELLS (COMPARE HUMAN OR ANIMAL CELLS TO PLANT CELLS)8) MOLECULES9) PROTOZOA10) VIRUSES(\QUESTIONS COULD COMPARE SMALL THINGS. E.G., HOW MANYCORPUSCLES WOULD FIT ACROSS THE DIAMETER OF A HAIR?MAKE A BULLETIN BOARD OF SMALL THINGS.(\158IDEA FROM:Arithmetical Excursions: An Enrichment of Elementary MathematicsPermission to use granted by Dover Publications, Inc.

Materials Needed:Centimetre cubesActivity:1) Arrange 4 cubes in each of the ways shown below.a) b)crrro./ ./ ./ .//' ./,/,c) d)./ ./ /'/' ./'./ /'./'./'2) Use your models or the diagrams above to fill in the table.SURFACE AREA VOLUMEMODEL It\) CJm.'.? It-.! C/ffi3abcd3) If the mass of 1 cube is1 gram, what is the massof each model?a)b)c)d)4) Does the mass of each model depend upon its surface area or volume?5) Does the amount of paint needed to cover each model depend upon its surface areaor volume?IDEA FROM:Geometry in Modules, Book B, by Muriel Lange,Copyright © 1975 by Addison-Wesley Publishing Company, Inc.All rights reserved. Reprinted by permission.159

A fly can sit on the surface of water.A fly covered by water is helpless.(\A person will usually sink.A person covered by water hardly notices it.a) Does the mass of a fly or a person depend on volume or surface area?b) Does the amount of water covering a fly or a person depend on volume or surface area?mine the number of cubic centimetres of water on the fly and on the person.c) The water covering the fly or person is about .05 cm thick. Use the table to detercMASSVOLUME (el'\'\'3) SURFACE AREA (em")fly_F'LY .019PERSOM ,0 k'j.04 .G"+580,000 9

Materials Needed:Centimetre cubesActivity:1) Build each of the models shown below.2) a) Which model is the "smoothest?"b) Which model is the "bumpiest?"3) Use your models or the diagrams above to fill in the table.SURF"ACE VOLUMEt-tIODE.L AREA (YIT{l.) (CJ'(Y\3)a.bcd.4) Does the "bumpiness" of amodel affect the surfacearea or volume of themodel?5) If each of the models abovewas an animal, which one wouldneed the most oxygen?6) If each of the models was an animal that breathed oxygen through its skin, whichhas the most skin to take in the most oxygen?IDEA FROM:Geometrv in Modules, Book B, by Muriel Lange,Copyright © 1975 by Addison-Wesley Publishing Company, Inc.All rights reserved. Reprinted by permission. 161

( ,An earthworm absorbs oxygen for its cells through its skin.If a large earthwormand a small earthworm have the same shape, which one do you think will be able tomore easily absorb the oxygen it needs?Suppose the earthworms shown have the dimensions given.4lU:lIbIbJI 1111 1II111!IllJJD,Length = 6 cmSurface = 12 cm2 ~L.\4!-1.JJ;>AreaVolume = 2 cm 31) a) The length of the large earthworm isearthworm.b) The surface area of the large earthworm isearthworm.c) The volume of the large earthworm is_____C___ times as long as the small--"--Length = 12 cmSurface = 48 cm2AreaVolume = 16 cm 3times that of the smalltimes that of the small earthworm.2) How many square centimetres of skin provide oxygen for each cubic centimetre of (volumea) for the small earthworm? b) for the large earthworm?3) Give the surface area and volume of an earthworm, similar in shape to the onesabove, that has a length of 18 centimetres.a) Surface area b) Volumec) How many square centimetres of skin provide oxygen for each cubic centimetreof volume?4) Which of the three earthworms would have the hardest time absorbing enough oxygen?5) Suppose there was a monster earthworm that was 100 times as long as the smallearthworm.a) Its surface acea would be ____C_ _b) Its volume would be ______C_'--'--____C_____C_'--'--~____C_ _c) How many square centimetres of skin would provide oxygen for each cubic unitof volume? (1626) Is it likely such a monster earthworm could exist?IDEA FROM:Geometry in Modules, Book B, by Muriel Lange,Copyright © 1975 by Addison-Wesley Publishing Company, Inc.All rights reserved. Reprinted by permission.

The physical traits that an individual has are inherited from his parents. Traitsare determined by a combination of gene groups--one gene group coming from each parent.Eye color can be used as an example. Each parent possesses a gene group with two partscalled alleles that are related to eye color. Biologists use an upper case B to indicatethe allele for brown eyes and a lower case b for the allele for blue eyes. For eyecolor the B allele is dominant over the b allele, so a combination bf Bb will producebrown eyes. The following shows the possible allele combinations and the color of eyesthat will be produced.BBbrown eyesbbblu.eevesBbbrown eyesbBbrown evesThese last two (Bb and bB) indicate the same combination. Biologists usually writethe upper case letter first. BB and bb are called pure combinations because both allelesof the combination are the same. Bb is called a hybrid combination.The combinations that are possible are often shown by using a grid.example one parent has pure brown eyes and the other has pure blue eyes.For this firstPURl::BBROWtVBb Bb Bb ALL OFFSPRllJG WILL HAVE Bb COMBIr.JATIOIJ \tJ>lICHPURt: BLUE. PRODUCES BROWI\) EVES.b Bb BbNow consider two 'parents with hybrid brown eyes.163

(CO~TIIJl>('O)(I4YBRID BROWt-.lBbB BBHYBRID BROWNb BbBbbbTI4E CHANCE OF AN O\cFSP!

ANALYZINGTRAITSThe following are some physical traits that can be observed. The first exampleillustrates some of the activities that can be done. Similar types of activitiescould be done with the remainder of the traits.Are you a taster or a nontaster? A harmless chemical called PTC* (phenylthiocarbimide)can be used to test this trait. A small strip of paper soaked in PTC canbe given to each student. Tasters will probably experience a bitter tast~whilenontasters will experience nothing. (It might be wise to have some candy availableto counteract the bitter taste. Let the nontasters have some, also.) Have studentstabulate the results and find the percent of the class that are tasters. About 70%of Caucasians are taster~ while about 90% of Blacks are tasters.Tasting is dominant (a Tt combination will be a taster) over nontasting, so thethree possible genotypes would be TT, Tt and tt, while the phenotypes would be tasters(TT or Tt) and nontasters (tt). Grids could be drawn to show some possibilities.Percents, ratios or fractions could be emphasized.T T T t TTTTTTTTTtHTtTttTtH100% tastersNV\I\50% genotype TT50% genotype Tt3:1 ratio oftasters tonontastersNV\I\1:2:1 ratio ofgenotypes TT,Tt, tt1 2 tasters1"2 nontastersNV\I\1"2 genotype Tt1"2 genotype tt*Add 650 mg of phenylthiocarbamide to 1 litre of water.Boil to dissolve thechemical. Allow the solution to cool. The chemical or prepared PTC strips can bepurchased from most suppliers of biological or chemical materials.The following is a tree diagram of several families.possible genotypes TT, Tt or tt.ADAPTED FROM:Children and Ancestors in Action Biology Series by Stanley L. Weinbergand Herbert J. Stoltz, Copyright © 1974 by Allyn and Bacon, Inc.Used by permission of Allyn and Bacon, Inc.Genetic Continuity, and Why You're You, Intermediate Science Curriculum Study.Permission to use granted by D. C. Heath and Company, and Silver Burdett GeneralLearning Corporation.For each, mark in the165

ANAlYZ\NG1RA\T£(PAGE: 2.)I(o -t-IIALEO-FEMALE:.TASTl:RNO!JTASTERTASTERTA5TE.R.tJONTA'e>TER.TASTERGROUPf'ERCEf-lTTASTERSPCRCEN,NOIU TASTER£>V. S. WHITEU.S. BLACKAFRICAIJ BLACK.ENGLI'OHCf1ltJE~t:Al"lERICI\IJE'i>KIIJlO11J1:>lAtJ,0919Go70949451:309t+30"4~lG"19(Other traits which could be studied include:a) attached earlobes--unattached earlobes.b) tongue rollers--nontongue rollers.ATTACHEDUNATTACHEDADAPTED FROM:166Children and Ancestors in Action Biology Series by Stanley L. Weinbergand Herbert J. Stoltz, Copyright © 1974 by Allyn and Bacon, Inc.Used by permission of Allyn and Bacon, Inc.Genetic Continuity, and Why You're You, Intermediate Science Curriculum Study.Permission to use granted by D. C. Heath and Company, and Silver Burdett GeneralLearning Corporation.(

ANALYZ1NGTRAITS(PAGE 3)c) hitchhiker's thumb--ability to bendtop part of thumb back to almost aright angle with rest of thumb.d) widow's peak--hair grows from apoint in the middle of the forehead.e) clockwise--counterclockwisedirection of hair whorl onthe back of the head.f) upper teeth gap--a small gapbetween the two front teeth.)Other examples can be found in most biology or genetics books. For each examplestudents could try to discover the dominant trait by investigating tree diagrams oftheir families.ADAPTED FROM:Children and Ancestors in Action BiologV Series by Stanley L. Weinbergand Herbert J. Stoltz, Copyright © 1974 by Allyn and Bacon, Inc.Used by permission of Allyn and Bacon, Inc.Genetic Continuity~ and Why You~re You~ intermediate Science Curriculum Study.Permission to use granted by D. C. Heath and Company, and Silver Burdett GeneralLearning Corporation.167

a mode! 50r!nhQfitanc8/(, ,For this activity students will need at least 20 centimetre squares from coloredplastic and 20 centimetre squares from clear plastic. Put each set into an envelope orsmall bag.The colored squares (let's use red) can represent the dominant color (R) of animaginary animal and the clear squares represent the recessive color (1'). To performa cross have a student draw a square from each bag. Record the allele combination.Put the two squares together and look towards a light source. Record the color seen.Repeat the drawing several times. The results should always be a Rr genotype and ared phenotype. R RThe grid looks like this. Students shouldrealize that this cross between a pure dominantRrtrait and a pure recessive trait yields 100%hybrid dominant traits.RY'To continue place an equal number of red and clear squares in each bag. This (simulates the allele combination for hybrid red color. Have students draw a squarefrom each bag and record the combination and color. By repeating the procedure manytimes the results should approximate those shown in the following grid.RrR RR RY'Y' Rr rrPHENOTYPE R.ATIO OF 3 REDS TO I W~nE.GENOTYPE R.ATIO OF I PURE: RED TO 2. l-\YBRIDRED TO I PURE CLEAR.168A similar type of simulation can be done using two coins and letting heads (H)represent the dominant color and tails (h) represent the recessive color.probable outcomes of tossing two coins is t HH, ~ Hh and t hh which agrees with theallele combination for a trait.IDEA FROM:Modern Life Science, by F. L. Fitzpatrick and J. W. Hole,Copyright © 1970 by Holt, Rinehart and Winston, PublishersPermission to use granted by Holt, Rinehart and Winston, PublishersThec·

WffAT TVPEAilE YOII ?Blood type falls into four categories AB, B, A and O. These letters refer to certainantigens that are attached to the cell walls of red blood cells. Two of these areantigen A and antigen B. A person with blood type AB has both antigen A and antigen B.Type B blood has antigen B; type A blood has antigen A; and type 0 blood has neitherantigen A nor antigen B. Antibodies are substances in blood plasma which react with theantigens to cause clotting. A method for determining blood type involves adding antibodyA and antibody B to two blood samples and then observing the clotting reactions that takeplace. The figures below illustrate the process.ANTIBODY AAlJTlBODY A A~TIBODV BAlJTIBODY A CAUSESCLOTTING SO BLOOD\5 TYPE A.At>JTIBODY B CAUSESCLOTTI1--lG 50 BLOODIS TYPE e,.80TH AtJTIBOD IE.~CLOTTING 50 BLOODTYPE IS AB.CAU5E/>JEITI-lER ANTIBODY CA\)SESCLOTTUJG SO BLooD ISTYPE O.It may be possible to get the necessary equipment for determinin~ the blood type ofthe students in your class from the school nurse, a high school biology teacher or thelocal Red Cross Bloodmobile off Lee. Parental permission should be obtained before doingthis activity. If your students find their blood types, determine the percent of theclass that is in each blood group. If possible, combine the data from several classes.The percents below show the percents for Caucasians in the United States. (Statisticsfrom Lane Memorial Blood Bank, Eugene, Oregon.) Students could research how the percentsvary in other racesAB - Lt% B-IO% A- 42%For transfusions it is important to know blood type. Type AB contains no antibodies;type B contains antibody A; type A contains antibody B; and type 0 containsboth antibody A and antibody B. If a person received a transfusion of the wrong type ofblood, the antibodies would react with the antigen to cause clotting which would clogthe capillaries and circulatory failure would occur. The following chart shows whichtypes of ~lood can be used for transfusions. Students may be dble to determine theresults from their knowledge of antigens and antibodies.TYPE CAN R.E.CE:IVE:AB8AoABB,ABA,ABA,B,AB,OTHUS TYPE AB is CALLED THEUN1VER'OAL DOl\lOR AND TYPE. 0IS CALLED THE lJN1VER'2>ALRECIPIENT.ADAPTED FROM: Children and Ancestors in Action Biologv Series by Stanley L. Weinberg andHerbert J. Stoltze. Copyright © 1974 by Allyn and Bacon, Inc.Used by permission of Allyn and Bacon, Inc.169

WItAT TVPEARE YOII ?(CONTINUE:O)(\Since blood type is an inherited trait, grids can be drawn to show the process.The genotypes are shown below with two examples of combinations that could occur.Notice that both antigen A and antigen B are dominant over the lack of antigens (type 0) •~ALLI;:LI;: )TYPE GENOT'!PE OMBINA1\OM A 0A-B ABA AA AO8 BB OR BO0 AO 00E)(AtJlPLE.. 1A AA oR. AO0 006 0A A6 AOo 60 00El

JUST LIKE PEAS'NA~ODThe science of genetics or inherited traits was first systematically studiedby a monk named Gregor Mendel in the 1850's and 1860's. He did his studies using thecommon garden pea. Mendel studied seven traits which are listed below.DOMI/IJAl\JT RE.CESSIVe Mendel first crossed a puredominant trait with a pure reces­I) SEED SI-jAPE: ROLJI\lD (g) WRllJK.LED (v-) sive trait. As the grid shows,2) SEED COLOR YELLOW (y) G.REEI\.\ (y)all offspring would be hybrid.3) SEEDCOAT COLOR COLORED (C) WHITE (c.)R R.'4-) POD SHAPE IN1CLATED (r) WRIIJ\

JUST LIKE PEASI~A~nD( COt-lTIt-JUED )Mendel extended his work to experimenting with crossings involving two traits.following grid shows a cross between pure round seeds and hybrid green pod color (RRGg)with pure wrinkled seeds and pure yellow pod color (rrgg).TheUsing one allele from eachtrait, the pairings from the first parent would be RG, Rg, RG and Rg. From the secondparent the pairings would all be rg.rgQG Qg RGRrGgQrGg RrggQ'fGg RrggRrgg QrGg RY"ggQrGg Ry-ggQrGg R.ygqRrGg Rrgg Q'fGg Qyg

C£OJSA2Yalleles. Parts of genes. One allelefrom each parent combines to determinea particular physical trait.The alleles that determine eye colorare usually designated as B forbrown and b for blue.alveoli.Tiny air sacs in the lungs.antibody. Substances in the bloodthat react with antigens to causethe blood to clot and to preventinfections.antigen. Bits of protein attached toblood cells.biorhythm. Theory that says a person I slife is explained in terms of threecycles: 1. physical (23 days) 2.emotional (28 days) and 3. intellec~tual (33 days).blood type.son has.The kind of blood a per­Can be 0, A, B or AB.calorie. 1. One calorie is theamount of heat needed to raise thetemperature of 1 gram of water byone Celsius degree (also calledsmall calorie). Symbol: cal2. (usually spelled with a capitalC) 1000 calories. (also calledlarge calorie). Symbol: kcal; Calcapillaries. Smallest part of thecirculatory system where the transferof oxygen from the blood to thebody and the transfer of waste productsfrom the body to the bloodtakes place.cubit. The length from the elbow tothe fingertip. About.5 metres.depth perception. Ability to judgethe distance between two objects.digit.middlethan 1As a measure the width of afinger, Usually a little morec.entimetre.dominant trait. When paired with arecessive trait, this is the traitthat will be seen in an offspring.In eye color brown (B) is dominantover blue (b). The combination Bbwill produce brown eyes.ESP. Extrasensory perception: transmissionor reception of stimuli thatoccurs without the use of the fivenormal senses of sight, hearing, smell,taste or touch.fathom. The distance from fingertip tofingertip when the arms are stretchedout in opposite directions. About 2metres.femur.genotype.nation.all haveintestines.digestingproducts.A bone in the upper leg.Fibonacci numbers. A sequence of numberswhere the next number is obtainedby adding the previous two numbers:1, 1, 2, 3, 5, 8, 13, 21, ...genetics. The science of studying theinherited traits and variations oforganisms.Refers to the allele combi­In eye color BB, Bb and bbdifferent genotypes.handedness. A person that uses hisright hand most of the time has righthandedness.humerus. A bone in the upper arm.Often called the funny bone.hybrid trait. A trait where the allelesare different, such as Bb.inherited traits. The characteristicsreceived from parents.Tubes in the abdomen forfood and eliminating wasteLilliputians. Tiny people written aboutby Jonathan Swift in Gulliver's Travels.all-HI 173

BIOLOGY GLOSSARYlun-villung capacity. Amount of air thelungs can hold.muscle fatigue. When muscles gettired from doing work.navel.A belly button.offspring. Children of humans (alsoof animals and plants).palm. The width across a hand, notincluding the thumb.peripheral vision. Ability to beaware of objects off to the sidewhile staring straight ahead.phenotype. Refers to the visibletrait. BB and Bb both producebrown eyes, so they have the samephenotype.PTC. Phenylthiocarbimide. A harmlesschemical used to distinguish tastersfrom nontasters, a trait inheritedfrom parents.pulse. Number of heartbeats perminute.pure trait. A trait where the allelesare the same, such as BB or bb •transfusion. Getting blood fromanother person.tree diagram.used to showpassed on toultimate height.will grow.A sketch that can behow physical traits areoffspring.How tall a personuniversal donor. Persons with bloodtype AB. Their blood can be givento persons with all other blood types.universal recipient.blood type O. Theyother blood types.Persons withcan receive allvilli. Tiny projections in the smallintestine that aid in digestion.J,.radius.A bone in the lower arm.reaction time. Amount of time neededto respond to a stimulus.recessive trait. Will be seen onlywhen a recessive allele is pairedwith another recessive allele. Ineye color blue (b) is recessive, sothe combination bb will produce blueeyes.span. The distance from the tip ofthe thumb to the tip of the littlefinger when a hand is spread out asfar as possible.tibia.A bone in the lower leg.174

to MATHEMATICS AND ENVIRONMENTEnvironmental problems have received much publicity in the past few years.The population increase and indifference to the dwindling of natural resources havecreated many environmental problems. Our students. must become aware of the challengesand changes necessary to give all people adequate food, clean air and water;appealing places to live and visit; and enough materials and energy to maintain ameaningful existence.~.,,. ':; :: :'( _. ,\",\VAW/\"'_/. - . .. .~.,How is mathematics used in environmental studies? In many ways. Indeed, it isdifficult to imagine a factual discussion about the environment that does not usemathematics in some way. Measurement techniques are used to determine the type andextent of water pollution; percents are used to contrast population densities; informationabout population sizes are reported with large numbers; order properties areused to compare populations; and word problems occur in studying the effects ofpests or contrasting the effects of pesticides. Statistical skills are essentialin environmental studies: it is necessary to read and interpret graphs and tablesof information; polling and other sampling techniques are used to obtain data; thedata obtained must be organized and displayed with diagrams and various graphs.Arithmetic skills, calculators and computers are often used in analyzing environmentalinformation. Problem-solving skills and critical thinking are needed toidentify various factors of an environmental problem; to decide on techniques toanalyze and solve the problem; and to critically interpret the data and claims ofpeople, organizations and governmental agencies.175

INTRODUCTIONENVIRONMENTThe pages in this unit can be chosen for use in a variety of ways: to reviewparticular mathematical skills, to tie mathematics into an environmental concern,to permit students some choice in the material studied, to support a short mathematics-science"mini-course" or simply to provide additional practice on some topic.You might prefer to substitute local or more recent data whenever they are available.The "Mathematics Topics" column in the Table of Contents for the student pages liststhe mathematics involved. This is usually the most advanced work on the page and isnot a description of the only topic used there.After some "getting started" activities, the first section deals with Population.Here you will find worksheets and suggested ideas that develop a sense of populationsizes, contrast in population distributions, and relationships between populationsand pollution. The next section on Food deals with interrelationships between livingthings; between food supplies and population growth; and with the effects of pestsand pesticides. The section on Air examines sources of air pollution; data relatingto air pollution; and alternate methods of transportation to reduce pollution fromautomobiles. In the section on Water you will find computational and graphing activitiesthat develop an awareness of water consumption and the necessity of water tosupport human life. The section on Land and Resources uses percents and rates to (study the uses of land and to develop an awareness of the magnitude of litter and ofthe limited resources we have abailable to us. The final section on Energy uses avariety of arithmetical skills to study different forms of energy expenditures.It is likely that the current generations of students will during their lifetimesbe involved in decisions which determine the ultimate quality and quantity oflife on earth. It seems appropriate that we should show our students how mathematicscan he utilized to help people make those decisions.(176

Students will often be more interested in a topic like pollution when it concernstheir immediate environment. The population of foreign countries will not beas meaningful to them as the population of their own city or town. If a nearbyriver is polluted or is becoming polluted, water pollution is a real concern.Start by gathering statistics about the immediate area, Have students helpyou find information from the following sources.Look in the Telephone Bookin the governmental listings under city, county, state and U.S.:Air Pollution AuthoritySanitation DepartmentWater Pollution BoardU.S. Forest ServiceSoil Conservation ServiceFish and Wildlife CommissionsParks and Recreation DepartmentTraffic EngineeringEnvironmental Quality DepartmentMotor Vehicles DepartmentBureau of Land Managementin the yellow pages:Recycling CentersWater Pollution ControlGarbage CollectionAir Pollution ControlEngineers: Environmental and AcousticalAssociations (environmental clubs in the area)Call or write and ask for free brochures, pamphlets or publications that includestatistics in any form,particularly numerical data, tables or charts.Read Newspapers and MagazinesCollect articles or advertisements that relate to pollution, to the energycrisis or to other environmental concerns. The information in these materials caneasily be used to make up mathematics questions on a student sheet. A sample ofsuch an advertisement is shown on the next page.177

(\What America is doing to reclaim energy and materials from municipal solid waste.ORIGIN RESOURCE RECOVERY DIVISION EP~A~ ~(REVISED 12/74)UJ:z; 2250H 200fx

Use Student Pages and Project IdeasSome specific student pages or projects have been written to show how theenvironment can serve as a theme in a mathematics classroom. Start with communityconcerns and information and adapt the resource ideas to the local environment. Abroader environmental picture will emerge as the projects get more involved, butstart with simple projects.One activity is to have students look for information in the sources alreadydiscussed. Doing this can be too time consuming for only one person so you mightwant to have small groups search together or to narrow the area to be explored.Statistics will differ, sometimes even with the same source at different times(many are based on estimates). Good information retrieval and processing skillswill be needed to obtain the most accurate information available and to interpretthe information correctly.If you plan to draw heavily on an environmental theme, you might wish to makea transparency of the People Need page or prepare a chalkboard outline. Studentsshould be asked to contribute their ideas. The major sub-themes of this unit-­Population (people), Food, Air, Water, Land and Resources, and Energy--should allbe listed. Suggestions like "houses," "recreation" or "sleep" should be retainedbut can be related to Land and Resources, Water or Air (noise pollution).If you compose a People Need list, you might invite students (beforehand?) towrite a description of things that could threaten the human race. Many ideas willstrike at items on a People Need list--and may come appallingly close to describingevents that are already taking place in at least a few locations on earth: starvation,disease spread through water supplies, poisoned air, eroded or sterile land,lakes and oceans threatened by sewage, oil spills and pesticides,Students could also contribute to a People Don't Need list. No doubt severalsuggestions will deal with aspects of the environment--excessive noise, litter,vermin, excessive packaging, polluted air or water, deteriorating buildings. Thesecould be categorized under air, water, etc., for our People Need list. You mightwish to tabulate the most common "don't needs" and use them as focal points forlessons.You might hand out Word Hunt for a homework assignment and use the environmentalwords found (and not found) by the students as a taking-off point.Or, you might use Environmental Scorecard early in the year, work with environmentalproblems during the year and then give the scorecard toward the end of theyear. 179

((,(180

Encircle all mathematical andhorizontal, vertical, or diagonal.you can identify 50 words or more,environmental symbols or words. Words may beSome words may be in reverse order. See ifother than those marked as examples.S E v E N u P E R c E N T I M EGDEN I LoDY GI V(T__D_~EEcoIpS I R A T SI N A T E AoEsI N EN H N U RP 0 HID C V P I SAO 0 L TRLAN D FILLLU C B DLCGI N H E DD M USQUARE 0EA V A E GU M ALUBANUNNL REM EJ I PIT E R MSSTI I B R YEARCIRe LEESoT 0T TFEA S R Q A K JH G ~R 0 D UoCLAT) UITRVAERUSDEE P SNKUNIAPRDNEDDA E RKSEICTuKEATANG E NTHToLIAPoLLUT I 0NNSNEoRSLLpEAG F 0xouDNoREIoGY R ATIoNoATATTyTI N USNGURcyILoAyLEeRUoSERAcoNENoI TeA R FEN UFINoEDAY (E__F__I__L__D L__I_i)SOU RCE: Pollution: Problems, Projects and Mathematics Exercises, 6 - 9, Wisconsin Department of PublicInstruction 181Permission to use granted by Wisconsin State Department of Public Instruction

~NVIQONMtNTSCOQtPut a check in the appropriate box.CRQO(Always Sometimes Never1) If I don't need to go too far, I walk or ride mybicycle. I don't ask to be taken in a car.2) I never throw trash on the ground. I put it in thenearest trash can.3) I don't make loud noise where it can bother otherpeople.4) I am careful not to waste water.5) I don't throw trash in lakes or streams.6) I help sort things from the trash that can berecycled.7) I am careful not to waste paper.8) I help keep our yard neat and clean.9) I pick up litter on my school yard.10) I let my mother know in advance when I need somethingwashed (like my gym suit).11) I don't waste electricity.12) I always clean up trash after a picnic or campout.13) I don't bother wild animals. I leave them and theirhomes alone.14) I take my own plate and silverware to picnics so Idon't waste paper and plastic products.15) I remind other people to protect the environment.16) I love this earth. I want it to be left nice forother people.(Scoring:For each "Always" check--3 points,for each lTSometimeslT check--2 points,for each "Never" check--l point.My environment score =Where can you improve your score?( ,182

[OMPUTCQC DNDTUb bNVIODNMbNTComputers can be used in different ways in the classroom, depending on studentbackground and access to the equipment. The study of environment can lead to twodifferent kinds of student activities on the computer: using simulations and writingprograms.Programming requires some understanding of a computer language while simulationsrequire none. They both enrich a student's learning experience.Simulation PackagesIf a computer terminal is available,find out what programs are already storedfor your use. An example of a simulationis one from the Huntington II projectcalled POLUT. The student controls thetype of water, water temperature, type ofwaste (industrial or sewage), waste dumpingrate and type of treatment. The outputis a graph or table showing the conditionscreated.Other Huntington II simulations concerning ecology deal with population growth,animal extinction and animal and insect control. The programs are written in a computerlanguage called BASIC and are short enough to be used in most school terminals.Each program can be purchased on paper tape and is accompanied by a student manual,teacher's manual and resource information. The entire package for each simulationcosts less than $3 and is excellent for anyone, even with no computer background.The Huntington II Computer Project materials are available from these two companies:Digital Equipment CorporationS"ftware Distribution Center146 Main StreetMaynard, MA 01754Hewlett Packard Company11000 Wolfe RoadCupertino, CA 95014Writing ProgramsThere are many activities on theenvironment that lend themselves to computerusage because of the complex calculationsinvolved. In some cases, handcalculators can also be used as a timesavingdevice and may be less expensivethan a computer. Some activities may bedone by the entire class, where each studentfinds different statistics but willuse the same formulas and program. Thecomputer would be quite useful then.183

[DMPUT~O~ DNDTW~~NVIODNM~NT(COIVTIIIlUED)One example of a student activity is about automobiles and air pollution. Thestudent is asked to find the amount of carbon monoxide in his residential districtat various times of the day. The student must find information on the number ofcars and the pollution production rate of each, then write the program, using givenformulas, to find the total amount of carbon monoxide. Additions to the originalprogram could be made to print out the concentration in parts per million, determinethe number of cars needed for the concentration to become lethal, change the site ofthe problem to two intersecting freeways, determine the effect wind has on the amountof pollution. The first activities do not require a sophisticated level of programmingskills and can be done by students with a brief introduction to a computer language.cMany similar activities are explained in detail in the Student Lab Booklet AirPollution and its companion Teacher's Advisor Book. Each book is available for approximately$1 from:Hewlett-Packard Computer Curriculum Project333 Logue AvenueMountain View, CA 94043Even if you do not have access to a computer, the booklets are an excellentsource for statistics, formulas, tables and charts. With the work divided amonggroup members, many of the activities can be done in the classroom.Listed below is a computer magazine that contains ideas for the mathematicsteacher. Suggest it to the school librarian.(Peoples Computer CompanyP.O. Box 310Menlo Park, CA 94025$5.00 a school yearComputer companies may also be able to furnish information about their role inenvironment or with environmental agencies. Write to the large companies to obtainmore information. (Students could also do this.)(184

COJIT&nJTSTITLE PAGE TOPIC MATH TYPEMATHEMATICS AND ENVIRONMENT: POPULATIONPopulationEstimation 189Developing sense ofrelative popUlation sizesFinding percents withlarge numbersWorksheetPeople-CountingRates 190Developing sense of sizeof Earth's popUlationCalculating with largewhole numbersWorksheetPicture TheseProportions 191Contrasing distributionof popUlation and naturalresourcesModeling percentsTeacher directedactivityYour Own SquareMetre 192Finding populationdensitiesUsing area, whole numberdivisionActivity cardPicture ThesePercents 193Contrasting popUlationdensitiesFinding, graphing percentsWorksheetDoubling 2, 4,8, 16 194Becoming aware of thepopulation growthAppreciating growththrough dOUblingTeacher ideasGraphing the World'sPopulation 195Becoming aware of thepopulation growthGraphingWorksheetHow Many Children? 196 Studying family size asa factor in popUlationgrowthStudying differentgrowth ratesWorksheetCounting Every Body 197Studying populationgrowth in U.S.A.Finding percentsTeacher pageWorksheetSizing Up the States 199Relating populationand pollutionGraphing and workingwith percentsWorksheetIf People WereLittler 201Seeing answers topopulation increasesUsing fractions in easyword problemsWorksheetPopulation Projects 202 Teacher ideasMATHEMATICS AND ENVIRONMENT; FOODLet's Go Counting 1 205 Finding population size Estimating population bysamplingManipulativeWorksheet185

(TITLEPAGETOPICMATHTYPELet's Go Counting 2206Finding population sizeEstimating populationby samplingManipulativeActivity' cardLet's Go Counting 3207Finding population sizeEstimating populationby samplingManipulativeActivity card208Studying the effects ofpestsSolving word problems withlarge numbersWorksheetFriend and Foe?209Contrasting the effects ofa pesticideSolving word problems,decimals, and percentsWorksheetFood for Thought210Relating food consumptionto population growthSolving word problems withpercen"tsWorksheetMATHEMATICS AND ENVIRONMENT: AIRParticularPollutants211Examining sources, kindsof air pollutionUsing decimals, percents,making circle graphsWorksheetAir PollutionCalling All Cars212213Seeking local data onair pollutionStudying cars as a sourceof pollutionPolling, graphing, usingpercentsTeacher pageProject(Up in Smoke214Considering the healtheffects of smokingUsing large numbers, readingbar graphsWorksheetWho Needs a BikePath?215Encouraging alternatetranspoFtation formsCollecting data, using ratiosInformation:PollutionAir216Studying facets of airpollutionUsing percentsWorksheetCi·ty Circumstances217Studying cities with airpollution problemsReading a tableWorksheetNoise218Background on decibelsTeacher pagePardon Me?219Studying the effect ofnoise on hearingCollecting, graphing dataActivityA Noisy Project220Studying the financial"tolerance for pollutionCollecting data, usingpercentProjectAir Projects221Teacher idea(186

TITLE PAGE TOPIC MATH TYPEMATHEMATICS AND ENVIRONMENT: WATERDo You Use MuchWater? 223 Developing awareness for Calculating with whole Worksheetwater consumptionnumbersDrinking, Hashing,and ? 224 Developing awareness for Calculating with whole Worksheetnecessity of waternumbersFix That Leak 225 Studying waste from even Determining rates Manipulativea small leakThe Wa"ter Cycle 226 Examining the water Using simple percents from Worksheetcyclea diagramWater Wastes 227 Studying the extent of Reading a "table, calculating Worksheetstream pollution in U.S.A. percentsWater Projects 228 Teacher ideaMATHEMATICS AND ENVIRONMENT: LAND AND RESOURCES915,000,000 Ha!231Studying the uses oflandUsing percents, large numbers,and circle graphsWorksheetLitter Counts232Comparing sources ofhighway litterFinding percent-fractionequivalentsWorksheetActivityLitter Watch233Developing an awarenssfor litterUsing whole numbers, tablesand graphsWorksheetActivityGarbage Grows234Developing an awarenessof size of solid wasteUsing decimalsWorksheetRecycling Cans235Encouraging recycling asa means of conservationReading graphs, percentsWorksheetRunning Out?236Developing awareness oflimited resourcesUsing percentWorksheetFour RecyclingActivities237Encouraging recyclingTeacher ideasA Grand CanyonEverywhere238Developing awareness ofeffects of erosionUsing large whole numbersWorksheetDid You Know?239Teacher pageLand and ResourcesProjects240Teacher idea187

(TITLE PAGE TOPIC MATH TYPEMATHEMATICS AND ENVIRONMENT: ENERGYGot Any Energy? 2'>1 Studying energy sources Using percents WorksheetHow Many Horses Illl 242 Developing awareness for Using a formula Activity cardssize of human powerSun Power 2L~3 Developing awareness of Solving a mix-ture of word Worksheetpotential sun power problemsPull for Pooling 244 Studying car pooling Taking a poll ActivitypotentialGet a Horse? 245 Comparing fuel efficiency Using decimals Worksheetof modes of travelFuel Facts 246 Studying factors that Using percents Worksheetaffect gas mileageBikes are Beautiful 247 Considering bicycles as a Using D=RT, percent Worksheettransportation modeEnergy Projec·ts 248 Teacher idea ((.188

PDPUL~TIDNb6TIMQTIDN1) Estimate the population of ..•D D D YOUR~D=~,-LL.::D=,----=D='-.J-L-'-L--' SCHOOL-----YOURClTY'

P~DPLb-[DUNTIN~ DQT~~'J1"-lE:'EEtU,TWEI\lT'I,(Needed:Watch or clock with second hand1) Keep track of how many seconds it takes you to countall the people in the room. Calculate your. number of people countedpeop1e count~ng rate = .. d .tlme 1n secon sHow long do you think you could count people at thatrate?2) Was your rate faster than one person per second? Even if it was faster, supposeyou counted one person every second. How many people would you count in oneminute? In one hour? In one day, without any rests?3) Suppose you could keep counting for a whole year. How many people could you count?(4) How many years would it take you to count all the people now in the world:4,000,000,000?5) Even if you lived that long, you would still be behind. Every day, 328,700 babiesare born and 131,500 people die. What is the daily increase in world population?At that rate how much would the population increase each year?At one person every second, could you count that many in ayear?6) In the U.S., 8800 babies are born and 5540 people die every day on the average. Inaddition, ~very year 395,000 people immigrate to (move into) the U.S. From thesefigures, what is the yearly increase in the U.S. population?Couldyou count that many in a year?7) What are some of the dangers of over-population? What can be done to avoid overpopulation?(190IDEA FROM: Environmental Science, Probing the Natural World, Level III, Intermediate ScienceCurricu lu m StudyPermission to use granted by Silver Burdett General Learning Corporation

The students build a model which shows the distribution of land, population and foodamong the continents.Let the classroom be the world.Land area1 1(or 25 or 30)desks so thatsizes. For example, North America has about15 of the world's land, so it would be representedby 4 desks grouped together.be obtained from Picture These Percents.PopulationEach desk could representof the world's land area.they are grouped by continentStudents stand or sit byData candesks in proportion to that continent's pop-1, h d ,11 .u at~on. Eac stu ent ~s 20' 25 or 30 aga~n(2~ is about 150 million people). With 2~being used as the fraction, North America(280 million people) would get 2 students.Food120MoveHave 25 penny candies, empty food containers or pictures of food and dividethem according to the part of the world's food supply that each continent produces:Latin America and Africa (1 each), Asia and Europe (8 each), North America (7).Use representations of other limited resources (e.g., oil) and try trading or borrowingto survive~Ask the students where they would liketo live or dislike to live, based on whatthe model shows. Discuss the implicationsof things like amount of land vs. population,population vs. amount of food, etc.IDEA FROM:Teaching Population ConceptsPermission to use granted by The Office of Environmental Education, Olympia, Washington191

Percent ofPopulation total Land area in Percent ofRegion in millions population million km 2 total areaUnited States 210 9.4Canada 20 10.0Mexico 50 2.0South & CentralAmerica260 18.5U.S.S.R. 250 22.4China 810 9.6India 570 3.3Japan 110 0.4Rest of Asia 710 14.3Europe 470 4.9Africa 370 30.3Australia + 20 8.5.TotalsData from Statistical Abstract of the United States 19751) Find the total population and the total land area. Record in the table.2) Find the percent of the total population each region has. Record.3) Find the percent of the total land area each country has. Record.4) Get or make a 10 by 10 grid. Show each region's percent of the total populationon the grid. Use different colors or shadings. Put the regions' names on theirparts.5) Get another 10 by 10 grid of the Same size. Show each region's percent of thetotal land area on the grid. Use the same code as you did in exercise 4.6) Compare the two grids. Which regions have more than their share of the people?193

DOUBLING 24816(When numbers keep doubling, they get very large after a few doubles. The population"bomb" is doubling faster and faster (see chart). But resources are dwindling, foodproduction is falling behind and technology is unable to provide "answers" like ediblealgae and solar power at a fast enough rate. Unfortunately, the doubling time is evenshorter in some of the poorest countries, leading to the likelihood of mass starvation.(It is estimated that 10,000 people die ofstarvation every day even now.) One groupWorldDoublinghas calculated that earth can support only Date population (est. ) time3.5 billion people at a reasonable standard1650 500 millionof livingl Although you may choose not todeliver such gruesome information to your 1830 1 billion180 yearsstudents, you can perhaps impress on them1930 2 billion100 yearsthe size to which doubling numbers canreach in a small number of doublings, and 1975 4 billion45 yearsthe necessity for slowing popplation growth. 2010(proj .)8 billion35 years1)1Clear the desks from about 4 of the room. Have 1 student stand in the clearedspace, then 2, then 4, then 8, etc. As students are added, have the class noticehow quickly each person has uncomfortably little room. (Adjust the amount ofcleared space if you have a small class or an unusually large room.) Since wehave only so much livable space on earth, the same overcrowding will eventuallyhappen if pur population keeps doubling.(2) (From The Limits to Growth, Meadows and others.The following is also shown in a movie of thesame title.) "Suppose you own a pond on which awater lily is growing. The lily plant doublesin size each day. If the lily were allowed togrow unchecked, it would completely cover thepond in 30 days, choking off the other forms oflife in the water. For a long time the lilyplant seems small, and so you decide not to worryabout cutting it back until it covers half thepond. On what day will that be? On the twenty-ninth day, of course. You have oneday to save your pond." Many students will think the 15th day. If your students arenot familiar with a lily plant, you might want to substitute something like weeds ina flower patch, junk on a playground or flies ("Kill a fly in May-you'll kill amillion a day. Kill a fly in June-not a day too soon. Kill a fly in July and youjust kill a fly"). A sequence of transparencies with dots for the lily plants (or (weeds, junk or flies) can make an effective demonstration.194 Permission to use granted by Universe Books

This table gives thepopulation of the worldfrom 4000 B.C.DateWorld population (estimated)Population (millions)1) How many years did it take thepopulation to double, from250 million to 500 million?500 million to 1 billion?1 billion to 2 billion?2 billion to 4 billion?Estimate when the population will bedouble the 1960 population.4000 B.C. 752000 B.C. 1500 2501650 A.D. 5001830 A.D. 10001930 A.D. 20001960 A.D. 30001970 A.D. 36001975 A.D. 40002) Use a whole piece of graph paper andgraph the ordered pairs in the table.Draw a smooth curve through thepoints.3) From the graph and exercise 1, what would be a reasonable estimate of the worldpopulation in the year 2000?4) Why did the population increase so slowly for so many years?IDEA FROM:Environmental Science, Probing the Natural World, Level III, Intermediate ScienceCurriculum StudyPermission to use granted by Silver Burdett General Learning Corporation195

(1) Some people don't have children. Some have 1 or 2 or 3 or more. How many childrendo you think you will have when you grow up?2) Suppose that on the average eachfamily has 3 children, each childhas 3 children when grown, and soon. How many children will be inthe 2nd generation?If the average keeps up, how manychildren will be in the 4thgeneration?CHILDREI\lTI-\£IRCHI LOR£t-lYOUo3) Suppose that On the average each family has 2 children in each generation. Howmany children will be in the 4th generation? (Hint: Make a diagram like the oneabove. )4) How many children will be in the 4th generation using your number from exercise I?5) Would a 4-child average givetwice as many children ineach generation as a 2-childaverage?6) According to the graph, aboutwhen was the population200 million?100 million?50 million?U.S. Population: 2· VS. 3·Child FamilyMillions~.~~_.--------fsoo--·_· _.--- ...• • •~.--------- !Ii1 !Ii!sooI -----.-1!Ii .-·------·_·--+·-i1700 ·--·~f~(1967) What could decide whether thepopulation is greater or lessthan 300 million by the year20007SOURCE:1870 80 90 1900 10 20 30 40 50 60 70 80 90 2000 10Commission on Population Growth and the American Future (19721, p. 23. Graph from Options;A Study Guide to Population and the American FuturePermission to use granted by the Population Reference Bureau, Inc.(

0CE) ~:;<

COUNTINGEVERY BODY219®oot;t-::W WZ 'd C9 CIlI I~ ::;!C/)@ :;:)(/)1-1. 2W uCENSUS ::>2«.-1 W:::! -2 u:z::I:(YQ!w C>!« aYEAR POPULATION gi?:Z2: V 1.

Choose your own colors.o -1 million1 - 2 million2 - 5 million5 - 10 million10 - 25 millionSr2II\r6UPTl-iE~TAT£~1) These population figures have been roundedto the nearest_2) Shade each state on the map on the followingpage according to its populationo4) The next 2 most populated states in orderareand5) These three states have what simple fractionalpart of the total U.S. population?8) Study the map you shaded. Why are some somuch more populated than others?Color usedIf you do not have a calculator, round thepopulation figures to the nearest half millionfor the percent problems that follow.3) The state with the largest population isIts populationis what percent of the U.S. totalpopulation?6) Find the fewest number of states required1to get 3 or 33% of the total U.S. population.List them.7) What is the fewest number of states thatcontain t or 50% of the U.S. population.List them.ALAHAMAALASKAARIZONAARKANSASCALIFORNIACOLORADOCONNECTICUTDELAWAREFLORIDAGEORGIAHAWAIIIDAHOILLINOISINDIANAIOWAKANSASKENTUCKYLOUISIANAMAINEMARYLANDMASSACHUSETTSMICHIGANMINNESOTAMISSISSIPPIMISSOURIMONTANANEBHASKANEVADANEW HAMSHIRENEW JERSEYNEW MEXICONEW YORKNORTH CAROLINANORTH DAKOTAOHIOOKLAHOMAOREGONPENNSYLVANIARHODE ISLANDSOUTH CAROLINASOUTH DAKOTATENNESSEETEXASUTAHVERMONTVIRGINIAWASHINGTONWEST VIRGINIAWISCONSINWYOMINGJuly 1, 1974PopulationEstimates*3,577,000337,0002,153,0002,062,00020,907,0002,496,0003,088,000573,0008,090,0004,882,000847,000799,00011,131,0005,330,0002,855,0002,270,0003,357,0003,764,0001,047,0004,094,0005,800,0009,098,0003,917,0002,324,0004,777,000735,0001,543,000573,000808,0007,330,0001,122,00018,111,0005,363,000637,00010,737,0002,709,0002,266,00011,835,000937,0002,784,000682,0004,129,00012,050,0001,173,000470,0004,908,0003,476,0001,791,0004,566,000359,0009) Which states probably have the most pol-TOTAL 210,669,000lution problems? The least? Why?*Statistical Abstract of the United States 1975199

(/... ....: ..,.. ..." tr ......I \\ \ I I\ \([. /\ I\,,, f\ II '" I\ (..l _.... ] ,~ __~~\ J \ ~ ../'- .....-, ) ,-, { "'-t..\. ... "'",,~ \ ... ,'"r ...', '",\ , 1\', () \ I ... ~ .. _ .......-... \~ -'"" I ~.........I .....') I I\ "I 1----......-1 ) \ I----,-1...., r----- __ ---1. I ~- '" ,/~ r..r-,./ -1--......... I.\.- ...... , ./1"- I ............ /-'r'" :--- I I I ... ~ __ ...I I _l_'"I 1 ...... ....,_J..----,...: )..""r' I (,..........,. J._ ........--- I I "_ ... - I" II l I I II l I I ,>I I I I II I I I rJ J I I r----i: : ,-_J : :I J ------;,--- L I I-- -,--_.J I ... .1 _---.., I ----.,, I '1 I' II I II : I, , ,I /__ ... I4-_ I -----,L _> ---,-..1 , ---- __II~J,I./.. _r.... I J___" ooJ I ,,......... I... _............. J --___ JI ' I - -../- ~I - ....-... I {, -..., :"~""''''' ... _.-,J I ,,/\ I ,I J ,;' ...( I //, ,- -- ... //-... I - __'"--\-.. IIII'"'"'"((200'"

IF PEOPLtW~Q~ LITTLtQDr. Igorovichsky has a solution to thepopulation explosion. He has invented amachine which shrinks people to half theirheight. Then people don't need as muchroom, they don't need as much food andenergy, and they don't pollute as much.Dr.Igorovichsky found these formulas:ISurface area of new body = 4 x surface area of old bodyIVolume of new body =8 x volume of old bodyIMass of new body = 8 x mass of old body1) Dr. Igorovichsky had a mass of 72 kg before he shrunk himself. What was his massafter shrinking?2) What would your height be after shrinking? Your mass?3) Dr. Igorovichsky ate 640 kilocalories each day after shrinking. How many kilocaloriesa day did he need before shrinking?to explain your thinking.4) Before shrinking, Dr. I. had enough food onIhand for 2 2 weeks. How long will the food lastthe shrunken Dr.I.?Be ready5) Dr. I's old car had a mass of 2000 kg. Now he needs a scaled-down model. Whatwill its mass be?Be ready to explain your thinking.6) Dr. I's pet dog was so excited at seeing his shrunkenmaster that he ran through the shrinking machine fivetimes. How does the dog's new height compare to hisold height?How do the dog's new and old volumes compareY7) Space scientists were very interested in Dr. I's machine. If a space lab couldhold 16 ordinary-sized people, how many shrunken people could it hold?Be ready to explain your thinking.8) What changes in the world would be necessary if there actually was a shrinkingmachine and everyone went through it? Would you go if everyone else did?9) What would be some other ways to cut down our consumption of food and goods?201

(1. Find which city has the fastest growth rate in your county. The office whichhas this information will vary from state to state. Try the records office orthe county information number. The local Chamber of Commerce might have figuresfor a few years. List some cities and their population changes in the past fewyears. Which city had the greatest increase in numbers? What is the growthrate of the closest large city? The greatest percent of increase? Look in analmanac and find the state that is growing at the greatest percent of increase.2. Find out how many classmates have pets. How would you estimate the number ofdogs or cats in your city?3. Look at a map of your state. Is one county of the state more populous thananother? Figure the population density (ratio of number of people to number ofsquare kilometres) of each county. Which is the most dense? Make up a colorcode and color the counties of the state on a map according to their populationdensities.4. Look in an almanac to find which state's population is most dense. Which countryis most dense?5. Use an almanac to find the country from which the greatest number of immigrantscame to the U.S. in a recent year (not in all almanacs). Or, graph the number ofimmigrants to the U.S. for a large number of years and try to explain any changes.6. Graph your school district's enrollment forable. (Call the superintendent's office.)to the city's population growth?the years that information is avail­Is the enrollment growth proportional(7. Find information from state tourist bureaus or your local Chamber of Commerceabout the number of tourists visiting your state or your city. What are the advantagesand disadvantages of tourism?8. Contact the motor vehicles department and ask for the out of state registrationsturned in last year. These figures will give you an approximate number of familiesmoving into your state. Compare them by state of origin. Are the majoritycoming from one state? Graph the results.The following are two examples of mathematics questions based on environmental data.9. In 1975 a whale was killed every 15 minutes on the average. How many were killedthat year?Japan has the largest whaling boats and killed about 210,000 out of the 550,000whales killed from 1960 to 1970 (figures rounded to the nearest 10,000). Whatpercent did they kill?Russia killed about 180,000 in the same period of time. What percent did Japanand Russia kill together? Why do they kill so many whales?The U.S. killed about 2,000. What percent is this?(Data from Project Jonah)(202Permission to use granted by Project Jonah

(p [ill ~ [ill [b [mu0[ill ~[p~[Q]JJ~IT:u~( COl-lTI t-lU£D)10. PESTICIDE USAGE AND AGRICULTURAL YIELDSPesticide UseYieldGrams perKilograms perArea or Nation hectare Rank hectare RankJapan 10,790 1 5,480 1Europe 1,870 2 3,430 2United States 1,490 3 2,600 3Latin America 220 4 1,970 4Australia and 198 5 1,570 5Pacific islandsIndia 149 6 820 7Africa 127 7 1,210 6Table from Patterns and Perspectives in Environmental Science, National ScienceBoard. A hectare is 10,000 square metres.a) Does a higher pesticide usage give a higher agricultural yield?b) Japan produces twice as much food per hectare as the U.S. but uses how many timesmore pesticide per hectare?c) The U.S. produces twice the agricultural yield per hectare as Africa but uses howmany times as much pesticide per hectare?d) What else besides the use of pesticides is related to the food yield?11. Students might be interested in writing poems about the environment. One student'spoem is below.PopulationThere once was a planet called earthIt was overpopulated by birthOne day it explodedAnd the people were corrodedAnd that was the end of the earth.203

c((

"City Has A Million Rats""2,500,000 Pines in Willumlaw Forest"How are those facts useful?How do they make such estimates?Are the numbers in these headlines exact or estimates?This page gives one possible way.1) Cut out a square region 1 cm on a side.2) Without looking, drop the square on the "city" (or "forest" or "lake") below andtrace around it. One example is already drawn. Count how many "rats" (or "trees"or "fish") are inside the square (4 in the example). This gives a sample. If thesquare does not land completely inside, drop it again.line, flip a coin to see whether to count it.If a dot is right on the"', ... ~..~3) Record in the table and tind 7 more samples.4) Find the average number (arithmetic mean) forthe 8 samples. This gives the average numberfor 1 square centimetre.5) Now find the area of the entire "city" abovein square centimetres.6) Multiply your answers in exercises 4 and 5to get an estimate of the total population.7) Count the dots to see how close your estimateis. How could you improve your estimate?SampleABCDEFGHNumber inSampleIDEA FROM:Junior Biologv: PopulationsPermission to use granted by the Board of Education of the City of Hamilton, Ontario205

(Sometimes wildlife specialists want toknow what percent of the fish in a bodyof water are game fish. Here is one wayto estimate the percents.Goal:To estimate the percent of eachtype in a mixtureoNeeded:Bag with 3 kinds of itemsIWithout looking, mix up the contentsof the bag.IIWithout looking reach in the bag andget a handful. Count the number ofeach type and record in a table likethat to the right. Put the handfulback in the bag.Sample 1Sample 2TypeA B C(Sample 3III In the same way, repeat I and II toget two more samples. Each timerecord and put the sample back.Totals% of grandtotalGrandtotalIVFind the totals for each type and thegrand total.VFind the percent each type is of the grand total.VI Count the items in the bag--each type and all together. Find percents. How closewere your estimated percents in V?(206

oSome chemicals do a good job of killingmosquitoes, flies and insects that eat crops.Chemicals that do not lose their strength veryfast can wash into streams and collect inbodies of water. There the chemicals arepassed along to plants, to fish that eat theplants and then to animals that eat thosefish.CONCENTRATION OF DDTIN ONE FOOD CHAINWaterBottom mudFairy shrimpCoho salmon,lake troutHerring gullsDDT in partsper million (ppm)0.0000020.0140.4103-699Data from Patterns and Perspectives in Environmental Science, National Science Board1) Why do the gulls have such a high concentration of DDT?2) Fish with the concentration of DDT given in the chart cannot reproduce. If a cohosalmon has a mass of 4.2 kilograms, how many grams of DDT does it have? (Assume aconcentration of 3 ppm.)3) Even penguins in the Antaractic have DDT in their bodies. The concentration is about0.18 parts per million. No one sprays DDT in the Antarctic. Where do the penguinsget it?If a penguin weighs 35 kilograms,how many grams of DDT does it have?4) Some bald eagle eggs have been found to have DDT concentrations from 1.1 to 5.6 ppm.Some osprey eggs have concentrations of 6.5 ppm. Why might osprey eggs have moreDDT? (Hint: What do ospreys eat?)5) Mosquitoes can carry malaria. At one time 3.5 million people died each year frommalaria. In 1966 there were 32% as many deaths from malaria as that. How manypeople died from malaria in 1966?Why had the number of deathsdecreased?209

(1) Farmers in the U.S. have been doing an outstanding job. In 1930 each farmer producedenough food for 10 others. In 1970, each farmer produced enough for 47 others.What percent of the 1930 figure was the 1970 figure?The average consumption of meats and2)Pounds % of totalfish in the U.S. is given to theBeef and mutton 120Poultry 50Pork 67Fish 13Totalcce"ecleft. Change each amount into apercent of the total. Is the averageabout like your eating habits?3) The average lower-budget 4-person family spent $2440 for food in 1973. In 1974,they spent $2730. What percent increase was that over the 1973 cost?4) To the right are given the averageamounts spent on some (not all) ofthe items bought in grocery stores,If the total bill was $20.00, whatamount, to the nearest cent, wasspent on each of the items?5) Add up your family's food bills fora week for each item in the list.See if your family spends about thesame percents as the average familyon the items on the chart.Item % of total Average costMeat ', •• , ., C 22.3Produce 11.0Dairy 7.1(inc1. eggs)Canned goods 5.8Frozen foods 5.2Bakery 3.8Candy, gum, 2.5cookiesSoft drinks 2.2Cereal 1.2Sugar 0.8( ,6) It is estimated that to keep up with the world population, food production willhave to increase by 20% to 35% eachyear. To the right are the figuresMILLIONS OF METRIC TONNESfor 1972 and 1973 for four important 1972 1973 % increasefood items. Is the food production Wheat 347 376keeping up with the population? Rice 294 322Corn 304 311Meat 88.1 88.2(210Data from Statistical Abstract ofthe United States 1975

PAIZTfCULA£POLLUTANTSTl-\(:Y USED TO HAVE '" '" = 0 0OXYGEI\l ltV THEIR "tR. .-" ~: I. ..-:~ I ,SOURCES OF U.S. AIR POLLUTION (1974, est.)•(millions of metric tonnes)Kind of PollutantCarbon Sulfur Hydro- Nitrogen Particu- TotalSource monoxide oxides carbons oxides lates for sourceTransportation 73.5 .8 12.8 10.7 1.3Heating .6 5.6 1.6 4.0 2.6Power plants .3 18.7 0.1 7.0 3.3Industry 12.7 6.2 3.1 0.6 n.oBurning waste 2.4 0 0.6 0.1 0.5Uncontrolledand misc.Totals5.1 0.1 12.20.1 0.8Data from Statistical Abstract of the United States 19751) Find the totals for each source and for each kind of pollutant.?) What was the grand total of air pollution in 1974?3) About what percent of the total pollution does transportation put in the air?4) Make a circle graph for either the sources (use the totals column) or the kindsof pollutants (use the bottom row). (Round the grand total to 200 million andeach of the other totals to the nearest million.)5) Transportation puts in lots more hydrocarbons than sulfur oxides. Power plants dothe reverse. Why is this?211

(Write or call the nearest air pollution authority. You should be able to find anumber in the governmental section of the telephone book. Ask for data in table orgraph form that relates to air pollution in your area.This is an example of a typical chart:SUMMARYSUSPENDED PARTICULATE1972Pollution Indexo -60 Light61 - 100 Moderate101 - 260 Heavy261 - Very Heavy-----City Station Location Field Station Suspended Particulate (J.lg/m 3 »,< Monthly Average YearlyNumberAverage- -Jan Feb Mar Apr May June July Aug Sep Oct Nov DecEugene City Hall 20-18-32 88 108 72 78 103 92 164 135 159 107 64 96 106COlrunerCe Bldg. 20-18-35 - 79 58 77 84 91 75 97 92 98 50 88 81Springfield City Library 20-33-37 114 138 92 89 93 59 88 107 109 142 75 95 100(City Shops 20-33-35 80 99 88 75 111 119 122 138 114 131 80 92 104Junction City City Offices 20-24-0L1 68 75 58 49 68 60 68 75 91 84 54 44 66Cottage Grove City Hall 20-09-01 - - 47 39 1,6 38 45 51 47 59 45 53 47Oakridge Fire Station 20-30-01 - - 85 56 89 78 79 105 83 106 73 69 82Rural LRAPA, Airport 20-00-33 25 36 25 23 38 48 42 50 53 72 30 32 40"---*micrograms per cubic metreThe amounts for each city could be graphed and compared.What statements can be made about the graphs?What are the sources of the particulates?What other pollutants are important locally?(Answers available from your air pollution authority.)(212

[RLLINCRLL [ARSToo many cars can cause air pollution.1) Take a poll among the students in your class. Find out how many families havea) no cars or trucks c) 2b) 1d) 3 or more2) Make some kind of graph for the information. Here are some samples.HOW MAt0'1VE\-\ICLES0: C:J 01: C::J C:J G G G2.: G C:J OJ3 OR. MORI:.: 0C:J • 5 rAMILlE':>

Suppose you, or someone you know, is one of the more than 70 million cigarettesmokers in the country today. Have you ever considered the following questions?(1. How many cigarettes will you (or a person you know) smoke during the remainderof your lifetime?2. What will the total cost of these cigarettes be?3. If all these cigarettes were placed end to end in a line, howlong would the line be?To answer these questions you will need the following information:a. The number of cigarettes on the average you (or the personyou know) smoke a day.b. The number of years you can expect to live.c. The cost of a pack of cigarettes.fd. The length (in millimetres) of one cigarette. : .,.'.. . ',7.0RATES OF CHRONIC BRONCHITIS AND EMPHYSEMAFOR SMOKERS AND NON-SMOKERS6.0w z0w 5.0 ~w~• MALE!ilffi'!IFEMALE0Sl4.0 ~w~wti~3.00 we-w :02~2.0'" « 1.0oFORMERSMOKERS(heavy andlight smokers)UNDER 11 11-20 21 AND OVERPRESENT SMOKERS BY NUMBER OFCJGARETTES SMOKEO PER DAY-HEAVIEST AMOUNTGraph from Patterns and Perspectives in EnvironmentalScience, National Science Board214

Do a lot of cars and bikes travel nearyour school each day? Many cities areestablishing bike paths so that bothtypes of vehicles won't have to travelin the same lanes.Find several locations to take bike and carcounts, and collect the following information.Street Time Bikes/hour Cars/hourBike approximateto car ratio----'--------------------------8-9 AVi G4 252.Do you think a separate bike path is needed?tion so that a path could be established?If so, who should have the informa~Do you think the number of bicycles would increase if a path were established?Why should bicycles be encouraged?Compare the motor vehicle registration and bicycle registration for your city.Look at each for the last 5 years to see if there are any general patterns.215

))JFO~MA1[ON:dee poccurlON ( ,1) Every year, about 180 million metric tonnes of pollution goes into the air. Wh~tpart of a metric tonne is this for every person in the United States? (Assume thatthere are 215,000,000 people in the U.S.)2) About 85% of the air pollution is invisible.How many tonnes a year are invisible?3) It is estimated that air pollution costs $65per year for each person because of propertydamage, upkeep, repair and cleaning. Whatis the total U.S. air pollution expense peryear from these causes?L C5W\-\O, ME?JO~OAbout half the air pollution is caused by motor vehicles.AUTOMOBILE POLLUTANT EMISSIONS(grams/km traveled)1957- 1970- 1972- 1975 temporary1967 1971 1974 standardsPollutant average figures standards U. S. Calif .Carbon monoxide 54 21 17 9.3 5.6Hydrocarbons 5.4 2.5 1.8 0.93 0.56Nitrogen oxides n.a.* 2.4 1.9 ('73) 1.9 1. 24~E.WER KIr0D :CAR FUt-I\E.S + SUN :PI-\OTOCI-IEMICALSIv\QG(*not availableData from Statistical Abstract of theUnited States 19754) Why does California allow less pollutants than the U.S. standards?5) The 1975 legal standards were 10% of the 1970 figures. How did the 1975 temporarystandards compare with the 1975 legal standards?6) Find out how many kilometres your family car is driven each week.above, calculate how many grams of air pollution it probably putsEach year.7) There are two kinds of smog.OLOER KIt-lD: SMOKE. FOG = SMOG"-t~.:::~.'-·::·:>~)\ ~'.:':I':':' ....With the chartout each week.216Is either kind a problem where you live?

_28CITYCIRCUM5TRNCE5SGSPENDED PAR'l'lCl.'I".\TE J\L'l.'1'Tl':n LEVl::LS, SEU;C'l'ED CITIES: 1972(In micrograms per cubic meter. JI$TATIOK :i\1!lL ~rax. GCOll1. STATWl\ Mill. 11a:", GeolU.meanmean----Arizona._ .... .. Phoenix__ .... __ 60 :':83 144 NewCalifornia.• __ .. OakJand _____ .. 21 100 57 Hampshire____ Concord _______ 20 93 38San Diego....._ 26 I 1'"L.... n"tf"d"''''. 25 137 60 Columbus. __ •.. 28 138 78New Ha,(.'n.u_ 23 131 60Oklahoma._.,. __ OklahomaFlorida.w. ___ .• MiamLh.... __ . 32 13:l 62 CiLy ___ .... 2S 19·1 67GO"g;L....... AtI,ntL"."' 1 30 :;03 821HawaiLm,.,n. Honolulu._. __ . 25 ;0 40 Oregon. _________ portland_______ 29 255 86Idaho ___ . _______ Boisem _. __ .. __ 20 I :;54 60 Pennsylvania___ Pittsburgh ___ .. 60 259 135Illinois_. ________ Chicago ________1" 336 97 Rhode Island. __ Providence_____ 38 129 69Iowa_______ .'_. Des ),10ine5_n_1 31 :;02 85 BOllth Carolina_. Columbia.. ____ 23 157 60Tennessee... __ ._ ),femphis_______KansllS __________42 395 9ZKansas City___ 29 227 116 Nashvllle._~ ___ 249 106Louisiana_. __ .._ New Orleans.._ 43 11' 73 Texas•.. ___ . ____ Dallas..._. _____ 29 319 86:\farrln.nd __ ._.__ Baltimore_. __ ._ 44 186 00 HoustOll _______ 43 177 89t-Hchigan________ Detroit_____ n __41 236 III San Antonio.u HI 54Minnesota_______ Minneapolis__ ._ 31 130!'.1JssourL.__.. _._ Bt. Louis Utah._. ______"__ Salt Lake City_211 95(CAMP)l____ 32 3tl3 120 Virginia___ . "_. __ Norfolk ______ ._" 9 116 55Nebraskan_un Omaha_________ 43 331 113 Washington. __ .• Seattle•. _._. __ . 24 109 17Nevada••_..____ Reno___________38 218 03 ,West Virgillia.... Charleston_____ 33 181 98Wisconsin _______ Milwaukee __ . __ 41 197 9ZWyoming....... Cheyenne.•••• _ 7 14. 30- Reprosents zero.1 At Continuous Air Monitoring Project (CA:\lf') Station.Source; U,S. Environmental Protcetion Agency, .l!oniloring and Air Qu.ality Trends Reporl, lU72."Table from Statistical Abstract of the United States 1974(Suspended Particulate Matter: Particles of smoke, dust, fumes and droplets in the air.)1) Using a standard of 75 microgram!m 3 (annual geometric mean), what cities had too muchsuspended particulate matter?2) Which city had the highest geometric mean? the lowest?3) What city had the highest maximum amount of suspended particulate matter?4) These are selected cities, not the only cities with air problems.are not represented?How many states5) What are some cities with obvious air pollution problems that are not mentionedhere?6) Suspended particulate matter does not include carbon monoxide, ozone, sulfur oxidesor nitrogen oxides. What information do you need before you decide whether thecities in the table have polluted air?217

"' ':,~'~:,':>' ," ..~ '. ,": ~ ; ./ / ." ':,I I • "Sound Levels and Human ResponseHearingEffectsConversationalRelationships(Carrier DeckJet OperationJet Takeoll(200 feet)DiscothequeAuto Horn (3 teet)Riveting MachineJet TakaoH(2,000 leet)Garbage TruckN.Y. SubwayStationHeavy Truck(50 feet)Pneumatic Drill(50 feet)Alarm ClockFreight Train(50 feel)Freeway TraHie(50 teel)Air ConditioningUnit (20 leel)Light Auto Traffic(100 feet)Living RoomBedroomlibrarySoft Whisper(15 fe~t)BroadcastingStudioConversation, 12 fl.An introduction to decibelscould follow work with noise,particularly if you can borrow anoise level meter. Studentscould then experience soundswith their associated decibellevels. Have students find thelevels of noise around school;classroom, hallway, cafeteria,bandroom, library.Several activities on soundmay be found in the MATHEMATICSAND MUSIC unit in the resourceMathematics in Science andSociety.The decibel unit is namedafter Alexander Bell; a decibelis 0.1 of a bel. The definition (of the unit is too advanced for .most middle schoolers, but onepoint should be made: A decibellevel of 40, say, is not twiceas intense as a level of 20decibels; it is 100 times asintense. Since loud sounds areoften quite intense, it is perhapsnot too misleading to saythat the decibel level measuresthe loudness of sounds.Here are some other notes onsound and decibel levels:Chart from Noise Pollution, Environmental ProtectionQuiet(!) classroom--20; cityAgencytraffic--75; Air Force recommendedmaximum--82; inside acity bus--85; sports car--86; screaming child--92; pushing a lawn mower--96;outboardmotor--l02; motorcycle--lll; kills mice--160; rocket engine--180. Eardrums can break atfrom 120-130 decibels.218Noise can make it hard to think. Some schools near airports have had to close becauseof this. Some office buildings in large cities use sound-proof glass to reducenoise. Besides causing hearing loss, noise can increase the stress on people, make themshort-tempered and even lead to high blood pressure. It is estimated that some workers (spend 15% of their energy "fighting" noise; some jobs require ear plugs. Students may .know that loud rock music has damaged many young people's ears. Students could be askedto suggest ways to reduce noise pollution.

,/Needed:Watch, radio, metre stickCan listening to loud noise affect your hearing?1) Have someone hold a metre stick nextto your ear (not touching the ear).Move the watch closer to your earuntil you can hear it tick. Recordthe distance from your ear.2) Sit close to a radio and turn it upvery loud. Do not turn it up soloud that it hurts your ear. (Getpermission from your teacher orparents.) Listen to the radio for10 minutes. Turn the radio off andrepeat step 1.3) Repeat step 2 the next day but listen tothe radio for only 5 minutes.~VDAY1C)p.'DAY 2-4) Repeat step 2 the third day but listenfor only 1 minute.C!}"5) Graph your results.6) What do you think might happen if you listened to loud noise for an hour? A day?A month?219

,/(Make plans to take a poll of 50 people(or if more convenient two homes per pupiland plan so that there will not be duplicationof households). Use these questionsor develop a set of your own that might bemore relevant to your community.Would you be willing to pay 8% more for an air conditioner if it made less noise? Morespecifically would you pay $310 as compared to $288?Would you be willing to pay $99 for a less noisy vacuum cleaner if the same model butnoisier was available for $90?Would you be willing to pay $1.25 more for a less noisy hair dryer?(Would you be willing to pay $5.00 more for a less noisy lawn mower?Tabulate your results on a chart similar to the one shown here." YES lt "NO" "UNDECIDED"ITEM NO. % NO. % NO. %AirConditionerVacuumClcnnerIlnirDryerL8wnl'v'1owcr,IIIf you can come to any conclusions that might influence a manufacturer of one ofthese items, why not write to such a manufacturer explaining what you have done andoffer some concrete suggestions to him for future production plans. Wouldn't it beinteresting to find out if the company would respond to your suggestions?(220SOU RCE:Pollution Problems, Projects and Mathematics Exercises, Wisconsin Department of Public InstructionPermission to use granted by Wisconsin State Department of Public Instruction

1. There has been a concern by environmentalists about aerosol products (hairsprays, insecticides) containing fluorocarbons. Find information about thepossible health hazards and about any proven side effects.2. What is an air inversion? Why is it a health hazard?3. Is air pollution likely to be greater in winter or summer? Why?4. Why are unseen pollutants likely to be worse than the ones we see?5. How have the laws changed regarding air emission standards for automobilesduring the past 20 years?6. Take polls in your class to figure out the following.a) How many ride bikes to school, ride a bus, are driven by parents or walk?b) What effect does the weather have on bike riding? (Count bikes on rainydays, cold days and nice days.)7. What methods of transportation do the teachers use to get to school?8. Keep track of the sounds you hear for a morning or evening or weekend day.Tabulate them as pleasant, unpleasant, loud, soft. Which ones can be turnedoff or avoided--loud TV versus train or traffic noise, for example?9. How many cars are there in your city? (Contact the motor vehicles departmentfor an estimate-)How much pollution would these cars give off if all were driven at the sametime as close as possible on streets for an hour? Use the information below.The formula P = R x n tells us the rate of pollutants emitted (F) with a givenrate (R) (33 £/km or 76.6 cu. ft./mi. or 2710 g/h or 2.7 kg/h per car) timesthe number of cars (n), at 40 m.p.h. Find the total produced for an hour (ora kilometre or mile).10. Find the ratio of cars to people in your city. Find the ratio of cars to peoplein your state. Compare the two ratios.11. Take a survey by looking to see how many people are walking, biking, busing orriding in automobiles. Is mass transit needed in your city?12. The average person breathes about 16 kg of air eachdrink he consumes). When you breathe polluted air,rise as well as bronchitis, lung cancer and colds.air pollution with deaths or disease.day (6 times the food andthe chances for emphysemaFind research to correlate13. Find statistics about production and effects of pesticides in the U.S.14. Graph daily pollution data from the local newspapers (usually in che weatherreport section). Such data may not be available in small communities.221

(\((

MY WATER USE (one day)1) On the average, how manylitres of water do youthink you use each day?Think of a typical day,or keep track for a day,and use the chart to theright to help you estimate.UseTaking bath 110shower 75Flushing toilet 22Washing hands,faceAverageamount (litres)Getting a drink 1Brushing teeth 17How manytimesTotal2) On the average, each personuses more than 200 litresof water at home every day:How does that compare withyour estimate?Doing dishes(one meal)30Washing car 40Cooking a meal 18Using clotheswasher90Total3) If everyone in the U.S. (215,000,000 people) took a bath every day, we would use acertain amount of water. How much water would the U.S. save each day if everyonetook a shower instead?4) What would be some ways of saving water at home?5) If you add in the water used in industry and agriculture, the average amount used byeach person is about 600 litres of water a day. From the information in numbers 2and 3, how many litres of water are used in homes every day?By industry and agriculture every day?IDEA FROM:McDonald's EcologV Action PackPermission to use granted by McDonald's Corporation223

Water is needed to grow food and to make almost everything we use.items and their average water "prices."Here are someOrSUNDAY NE:WS/ . -=-=-'" '.""-0......... ---How much water would be used ..•(1) if everyone in math class has a bicycle?2) for a year's supply of the "Sunday News"? (How many Sundays?)3) for the wheat in 10 loaves of bread?4) for the tires for all the family cars (remember the spares:)?5) for 3 cans of peas and 5 cans of beans?6) for 40 t of gas?7) for a dozen ears of corn?8) Why is so much water needed to make a bicycle?(224

FI X TfiAT L(AK~ t t ~ ~Materials Needed:Graduated beakerWatch with a second handCalculator, if availableActivity: (1) (a) Turn on a faucet so it drips at asteady rate.(b) Count the number of drops fallingfrom the faucet in 30 seconds.Repeat the count for accuracy.The water is dripping at a rate ofdrops per______ drops per hour.______ drops per day.(2) (a) Measure the volume of 100 drops of water in millilitres.(b) Use the data above. The faucet is dripping at a rateof _ millilitres per minute.millilitres per hour.millilitres per day.millilitres per year.rate is equivalent to a rate oflitres per year.__(3) Call your local water board to find the rate charged forresidential water use. (The rate will probably be dollarsper 1000 gallons of water. 1 litre = .2624 gallons.) Howmuch money is wasted by this dripping faucet in one year?225

.. .'., .· ',.;,' .(EVAPORATIOr-lt.18'YoOCEIWS At-lDSEAȘAdapted from Patterns and Perspectives inEnvironmental Science, National ScienceBoard;CLOUDSThe diagram shows the water cycle. The diagramdoes not show the water that is being stored orthat is in animals and plants.STREAMS \BO OI ~~ OF RA1 \-), Sr-lOI,JWA.TER. ~ I1) Evaporation turns water from a liquid to a vapor which then goes up to form clouds.What percent of the evaporation happens over oceans and seas?.. ~ 't / (PRE:CIP TATIOt-l)(2) Why don't the oceans and seas dry up?3) What percent of the rain and snow end up in the oceanS and seas?Out of every 500 litres of water from rain and snow4) how many litres end up in the oceans and seas?5) how many litres are evaporated on land?6) how many litres are evaporated from the oceans and seas?7)How much water (rain and snow) falls each year where you live? Look under "Precipitation"in an almanac for a place close to you.c226

WATER POLLUTION, MAJOR DRAINAGE AREASTotal lengthPolluted km1971 durationofstreams inName of area the area (km) 1970 1971 intensity factor*Ohio 46,400 15,800 38,500 .42Southeast 18,800 5,000 7,200 .74Great Lakes 34,200 10,500 14,000 .45Northeast 51,900 19,000 9,300 .61Middle Atlantic 51,100 7,400 9,000 .47California 45,200 8,600 13,500 .27Gulf 103,600 26,600 18,600 .35Missouri 16,700 6,800 2,900 .31Columbia 48,700 11,900 9,100 .12United States 416,600 111,600 122,100 :41*This indicates how badly the stream was polluted and for how long during the year.higher the factor, the worse the pollution.Table adapted from Statistical Abstract of the United States 1975The1) Which area within the U.S. has thegreatest number of kilometres ofstream length?2) Which areas had fewer polluted kilometresin 1971 than in 1970?3) Which area had the highest durationintensityfactor in 1971?The lowest?4) For each area, find what percent of thetotal stream length was polluted in1971. Arrange them in order, highestdown.--- '-.---. -.- .,---_...::-=-_.-.:... .-•-----~ - -- '----. ---=- ...........--...~(.\'1'\,1. \ II'--..---.,--- ;"-:--::... -;::::---:----5) What are some streams in your drainagearea?Are they polluted?227

(1. Is there water pollution in your neighborhood or school area? Look for informationin Consumer Reports for charts about detergents and their amount of phosphates.Select a group of private homes and interview people for the following information:brands and amount of detergents used per month. Figure the amount of phosphate fromcharts and estimate the amount of phosphate pollution for your city. Is it enoughto affect the water?2. Call the local water board to get the monthly use rates by the total city population.Use an appropriate graph to display the 12 months of information. It may be possibleto do a double graph, where both this year's and last year's use can be compared.3.4.If there is a water tower familiar tothe students, it is interesting to knowits capacity. Call the city engineeror the industry involved to find out 15its volume. How much water would each ~student get if the water in the tower ~were divided up? How many swimming jpools would the water in the tower fill? \.. 10A short activity like the followingmight lead to an exploration of localstreams, fish population data, watersources and sewage, methods of cleaningup polluted streams, •••o({l~ 5:3:fWAT£R COI-lSUMPTION IN 5OMETOWl\l(:rANFEBPOLLUTION-CAUSED FISH KILLS, 1973Source of pollutionFish killed(thousands)Insec.ticides, poisons,manure drainage2,500Industry 1,800Sewage, refuse disposal 10,400Transportation 600Unknown 22,600Data from Statistical Abstract ofthe United States 1975a. How many fish were killed by pollutionin 1973?b. Were any fish in your area killed bypollution?c. Many fish are killed by unknown pollution.Why is it so hard to trackdown the sources?(228

WRlERFigure VIII-14 - COMMERCIAL FISH CATCH: LAKE MICHIGAN100 1------------:/;=--2~~::.:=======:===~=1LAKE TROUTLAKE HERRINGSUCKERSWHITEFISH1) The increase of thetotal fish catch in1965-66 was due to anincrease catch of whatkinds of fish?YELLOW PERCHCARP"' ~ 40:>Ii'~ 30"' zo::;~ 20"SMELT1898- 1910- 1920- 1930- 1935- 1940- 1945- 1950- 1955- 1960 1961 1962 1963 1964 1965 19661909 1919 1929 1934 1939 1944 1949 1954 1959The diagram shows statistics of the commercial fish catch on Lake Michiganfrom 1898 to 1966. The degradation of the fish population is clearly evident;although the total catch returned to turn-of-the-century levels in 1966, almost allof it consisted of alewives and other low~value species.2) Fish catches were alldecreasing during whatyears?3) Find out what kinds offish are caught nearyou and how it haschanged in recent years.Graph from Patterns and Perspectives in EnvironmentalScience, National Science Board229

~ EEB""'-'~~wlci~X~;l\~J;;~i;lf,ki.;;i;;i:£i!JfjiP/l( I\\\~Here is how land in the United States is used.MillionsPercentUse of hectares of totalCrops 155Cities,transportation22Parks 251) Find the percent of thetotal for each use.2) Make a circle graph forthe information.Pasture 244Grazing land notin farms116Forests 213Other (desert,swamps, 140mountains)Total 915Data from Statistical Abstract of the United States 19752A hectare is 10,000 m3) Which kinds of land do you use?4) Suppose everyone had the same amount of land. How many hectares would you have tothe nearest .1 ha?kind of land would you pick?5) U.S. land is worth $715 billion dollarsmuch is a hectare of land worth?your share in exercise 4 be worth?(Assume there are 215 million Americans.)What(in 1968 dollars). On the average, howOn the average how much would6) Interstate highways use about 10 ha of land per km of highway. There are. ~bout68,500 km of interstate highway. About how much land does the interstate highwaysystem take?find the value of this land.Using the average value per hectare from exercise 5,7) It takes about .5 ha to grow the crops to feed the average American. How manypeople can be fed from the U.S. cropland?the surplus?What happens to231

Recent surveys continue to show that highway litter is composed of .••., .~ "True or1)2)3)4)false.1About 6 of the litter is cans.Paper gives about 10 times the amount of litter4Paper is about 5 of the total amount of litter.34" of the litter comes from paper and cans.as bottIes do.(iTake a litter count along the streets beside your school. Tally each piece into oneof the above types and change to percentages. How do they compare?If possible, find out information about litter along your state's highways. Has itbeen increasing or decreasing in recent years? How does it compare to this nationalsurvey?What can you do about litter?232Does your state require a deposit on every soft drink or beer bottle or can?information from a state that has such a law and a state that doesn't. Whichmore bottles and cans in the litter along the highways?Try to getstate ha~( ..

Students in one class at Washington School picked up and counted the litter at theirschool. Here are their findings.LocationPieces* of litterMTWThFWeektotals\tJAS\-lINGl"OfJGROUNDSSCI-\OOLNEast wing 8 8 12 11 10Central wing 25 15 14 15 23West wing 12 10 6 11 8School grounds 48 28 24 25 27Totalsw+s*A torn-up note or broken bottle was countedas one piece.SiREET1) Find the totals for each day and the weekly totals for each location. Record inthe table. Find the total number of pieces of litter for the week. Record.2) inside the school only.3) Graph the total number of pieces of litter day by day.4) Do you agree with this headline?Be ready to explain.5) There was a school assembly onFriday. In which wing do youthink the assembly hall isprobably located? Why?W[ST WJ~G:n-lE CLEANESTl\.v'ASI-lIl\lGTOt-J WEEIS]-=-- _._-6) On what day did the school grounds have the most litter? Why do you think thishappened?7) The Student Council decided to buy SOme litter cans. How many should they buy?Where should they put them?8) Plan a Litter. Watch for your school for one week.a) Who will be on watch-teams?b) Where will each team count the litter?c) When will they count? (Find out when the custodian cleans.)d) What will count as 1 piece of litter? (Is an ice cream wrapper and stick oneor two pieces of litter?)e) How will you keep other students from knowing that you are keeping count?f) How will you report your results to the school? 233

How much garbage, rubbish and junk do you throwaway each day?How could you find out? Try weighing the total amount thrownaway by your family for one week.(1) Each American produces about 2.5 kg of solid waste(garbage, rubbish and junk) a day, on the average.rate how much solid waste will be produced .•.by you in a year?by you since you've been born?by your family in a year?by your city in a day?2) If 1 cubic metre of solid waste has a mass of 90 kg, how many cubic metres of wasteis produced in your city each day?3) Measure your mathematics classroom and figure its volume in cubic metres. How many4)classrooms full of solid waste do the people in your city throwaway each day?How many days would it take you to fill the classroom at the 2.5 kgper day rate? (3Many cities pick up solid waste in compactor trucks that compress 3 m of waste into31 m. If this were done in your city, what would be the compressed volume of aday's waste?they compress the waste?Would this change the total mass?5) The "junk" bill in the U.S. is $17.50 per person per year. How much would this befor all the people in your city?Why do6) What is the most common method ofsolid waste disposal? Why is itused more than the others? Whichway is used in your community?Which way is best for the environment?DISPOSAL METHOD'S> - SOLID WAS,..£.OPEt-:l________77%DUMI='I~GIt-.lC.IN£.RATICM_ 8%SANITARY LANDnLL _13%COMPOSTIt-lC. OR SALVAG.I2.D .:2-%SOURCe - I-I.E:W. BURrAU OF 5Cl..\D WI\':>TE MAlJAG.EMa.lT7) If 3.5 billion metric tonnes of solid waste were discarded in 1 year, how manytonnes were left in open dumps?The local sanitation department should be able to give you figures for local daily or (monthly waste. Are there seasonal differences in waste amounts?234

Major efforts to recycle materialslike aluminum cans have begun inthe United States.RtCYCLINGCRNSUse the graphto help you with these questions.1) The number of aluminum cansAlUMllJUM CAN'::. RECYCLI:DI.e. ---------------1BILLlO\\)\.'2­BILLlOl\.lrecycled has (increaseddecreased)in the past years.770 --1't-JllLLlOW2) True-false: We probably recycled10 times the amount ofcans in 1974 than we did in1970.3) The increase in recycling from1972 to 1973 was what percentof the 1972 amount?197019711972.19734) If recyclers were paid $7 million dollars for their cans in 1973, what is theaverage price paid per can?5) If there were 48 billion aluminum cans made in 1973, what percent was recycled?Is it possible to figure out from the graph if a larger percent of thecans made were recycled in 1973 than in 1970?6) In early 1970 there were 100 collection points for recyclers, in 1971-~285, in1972--675, in 1973--1100. Predict the number for 1974 by drawing a graph or usingpercents of increase.7) How many students in your class do any recycling at home?Permission to use granted by The Aluminum Association, Inc.235

~\JNNING~V7?(Unless more is found,the known amounts of some resources will last this long:iron - 72 years copper - 38 yearszinc - 28 years lead 27 years1) If you live to be 75 years old, which of those resources are likely to be minedout?(Use the figures above.)To avoid shortages, we can use less of theresources and recycle them. Here are theamounts used and the percent recycled nowfor some resources.',:':'.~r'M~CO~SERVI~G!": .;I ',', '..... "../:.: .', ': ':(Resource Amount used/year % recycled Amount recycledCopper 1,450,000 t 60%Aluminum 4,450,000 t 48%Lead 610,000 t 42%Iron 89,250,000 t 26%Paper 61,100,000 t 21%2) Find the amounts of the resources that are recycled, in metric tonnes.3) Nine million cars are junked every year. If there are about 90 million cars inuse, about what percent are junked?4) Can all these resources be recycled: coal, oil, natural gas, gold, silver, wood?5) What can the students in your class do to help conserve resources.(236

l=OUQRl;[YCLING~[TIVITI{;51. Find out how materials other than paper are recycled.You might assign small groups of students tofind out what steps are involved in the recycling ofglass, aluminum, steel, and rubber. Have the groupsmake bulletin board displays, using a pie-shapeddesign, showing the steps involved.2. Recycle your own paper. Recycling paper commerciallyrequires huge machines and elaborateproduction lines. You can, however, demonstrate thebasic steps of paper recycling in your classroomorstudents can perform the same project at home.The materials you will need are: an old newspaper,mixing bowl, egg beater, a wood block, a square ofwindow screen (about3-4 inches on each side), aplastic sandwich bag, wallpaper paste or cornstarch, water and a tablespoon.1. Fill the bowl about one-quarterfull of water. Tear a half page ofnewspaper into tiny pieces. Placethe pieces in a bowl and let themsoak for one hour.2. After the paper has becomethoroughly soaked, beat it withthe egg beater. The action willbreak up the paper into fibers.When the mixture has been thoroughlybeaten, it should have thecreamy texture of paper pulp.3. Dissolve two heaping tablespoonsof wallpaper paste orcorn starch in a pint of water.Pour into the pulp. Stir. (Thepaste or starch acts as a binderbetween the wood fibers suspendedin the mixture.)4. Hold the piece of windowscreen flat and lower it into thepulp. Do this repeatedly until youaccumulate a layer of pulp about1/16th inch thick.5. Set the pUlp-covered screenon a newspaper and place aplastic sandwich bag over it.Press down with the wood block-gently at first, then with morepressure. The water will filterthrough the screen onto thenewspaper.6. Allow the fibers to dry for about 24 hours. Peel thefibers-now paper-from the screen. CongratulationslYou have just completed the process of recyclingpaper.3. Demonstrate nature's way of recycling. Nature hasits own way of disposing of some trash. You candemonstrate this process in your classroom. Theprocess takes two or three weeks to complete, soyou should allow for this length of time.You will need a large clay flower pot, top soil, afew small stones, and a collection of typical litterpiecesof newspaper, plastic, a strip of tin can (or asmall piece of steel wool without soap), broken glass,a few potato peels or pieces of bread.1. Cover the hole in the bottom of the pot with thestones. Fill about one-third full with topsoil.2. Add a layer of the litter you have collected.3. Cover the litter with a layer of top soli until the potis two-thirds full. Sprinkle with water, but don't soak.4. Cover the pot with a piece of plastic. Set it in awarm, dark pi ace. Keep the soil moist.5. After two or three weeks, dump the contents of thepot on a newspaper and with a trowel spread it out.What trash has decomposed? What buried litterdidn't decompose? What does this tell about theproblem of litter and trash? You might want to returnthe material to the pot and examine again in threeweeks to see what further decomposition has takenplace.4. Collect recyclable materials. On their own-or asa school project-students may want to collect trashthat can be recycled, such as paper. cans, bottles, orscrap metal. Before beginning a project like this,however, be sure you have a buyer-or recyclingcenter-to take the materials. Depending on thearea, aluminum cans sell for about $200 per ton,paper for $13 per ton, and glass bottles $13 to $20per ton.SOU RCE:McDonald's Ecology Action PackPermission to use granted by McDonald's Corporation 237

· ~.; '. '" ~ : .... (Erosion is natural.Humans increase erosion by plowing up land, removing plant coverand trees and allowing land to be overgrazed.3 B\LLIOl\l M£:TRIC. ,Ot-JIIlE,,:> OF SOtL ARE ERODEDEA.C.\-I yr.AR. T\-I15 COULD COVER GOO,OOO \-\EC'IIRE.SWIII-l 30 CEI'JilMETRE5 Or- ::'OIL.1) By the scientist's figures, how many tonnes of soil would there be for each hectarecovered?2) Each year the Mississippi River washes 450,000,000 metric tonnes of soil into theGulf of Mexico. According to your answer in exercise 1, how much land could this(cover with 30 cm of soil?3)2Up to 4500 metric tonnes of erosion can result from stripping 2.5 km of land of2plants. Lately about 1600 km of land have been exposed each year. How much soilcould erode from this exposed land?.'GRASSLAWD90kq/ha- EROSION4) Erosion in cornfields is how many times as much as erosion in grassland?5) How do farmers try to prevent erosion? Is there anything you can do?(238

DID YOUKNON?Here is a sample page you can make up from newspaper, magazine or brochure clippings.1970What America is doing to reclaim energy and materials from municipal solid waste.Encouraging news in the war on wasteORIGIN; RESOU(~~~lrig~~7~'; DIVISiON. EPA To date, more than 50 U.S. communities are planning or actuallyoperating resource recovery facilities. These facilities are designedto recover some of the $2 billion worth of reusable materialsin the estimated 150 million tons of refuse Americans willdiscard this year alone. Significant trends are emerging. The chartat left is based on a recent study by the Environmentai ProtectionAgency. It plots the expected growth of municipai solid waste (topline) ... and the projected increase in resource recovery (shadedarea). It shows that in 1971, we recovered only 6% (eight milliontons) of our solid wastes. By 1990, we should be recovering 26%(58 million tons). And, as the chart also shows, net waste disposalcould level off by 1985 as the increase in recovery equals the\975 1990 increase in gross discards.225f-----+---+----+---::&J"',1-'''''---+---+----+-----11) Given a population of 210 million people--how much refuse on the average will eachAmerican throwaway this year? each day of this year?2) We will discard how much more in 1980 than we did in 1974? What is the percent ofincrease?3) What was the percent of recovery in 1975?Did you know that ...The "garbage" Americans throwaway each year contains: enoughferrous metal to build 4 million cars ... enough paper to printevery major newspaper in the country for a year ... enough combustiblematerial to generate the energy of over 200 million barrelsof oil. 0 A recent study forecasts that 10% of America'sgarbage will be used for energy by 1980. Projections from anothsrstudy indicate that more than 300 communilies with populationsover 10,000 will launch major resource recovery projects in thenext five years. 0 Connecticut plans a network of nine regionalresource recovery plants, Contracts have been signed for the firstplant; it will provide the electrical energy needs for approximately10% of the Bridgeport region's population.7) In 1976, what percent of waste will be reclaimed in Milwaukee?8) How many tons will be reclaimed?Resource recovery, big city style: Milwaukee, Wise. (pop. 717,099)The lights afMilwaukee's skyline: next year some of them will belit with electricity generated by burning the city's garbage. Milwaukeehas become the first city of its size in the U.S. wilh a longtermcontract for systematic solid waste recycling. In mid-1976,an $18 million facility designed, constructed, financed and operatedby American Can Company's Americology unit is scheduledto come on stream. When it does, about 20% of Milwaukee'sannual 270,000 tons of municipal waste will be reclaimed for itsusable steel, tin, aluminum, paper and glass. Another 60% will beconverted into combustible material with the annual energy equivalentof 75,000 tons of coal. Wisconsin Electric Power Companywill buy this fuel to help light the city's homes and offices.4) In 1980, how many tons of garbagewill be produced total that yearby each American if the estimatedpopulation will be 220 millionfor that year? for each day?5) Is it more or less than theaverage per day for 1975? By howmuch?6) In 1980 will you throwaway moreor less than this year?ADVERTISEMENT FROM: Saturday Review, July 12, 1975Permission to use granted by American Can Company~'American Can Company,a, American Lane~....... Greenwich, CT 06830 239

L~ND ~NDR~~OURC~£PROJbCTS1) What methodes) does your city use to dispose of solid waste? What are the advantagesand disadvantages of each method?(2) Are there recycling centers nearby? Find out what objects and how many are beingrecycled each month, each year.3) Paper is a big part of solid waste and litter--both by count and mass.out how much your local newspaper contributes. What is the total massfamily's newspapers for 1 week, 1 month, 1 year? Find the circulationnewspaper and total production for a year (1000 kg = 1 metric tonne).paper printed on recycled paper?Figureof yourof theIs the4) How much paper does your school district use each year? Is it recycled paper?5) In what ways are our forests and national parks polluted?6) Find out how people feel about littering. Take a survey and see if anyone caughtlittering should pay a fine and/or spend at least one day picking up litter alongyour state's highways.7) Look up Garbage Collection or Services in the yellow pages of the phone book.Count the number that exist. Call some of them to find out what they think theaverage amount of solid waste is that's disposed of weekly by a family in yourcity. What amount do they handle in a week's time; in a year's time? Usingtheir figures and the number of families they collect from--estimate by propor- (tion the amount of solid waste collected in a week in the entire city.8) Find out from school cooks how much uneaten food isaverage. Compute this amount for the school year.waste discarded daily or weekly per student, yearlyreduced?disposed of daily as anFind out the total foodper student. How can this be9) Pick out 20 items in a grocery or drug store. See what percent has unnecessarypackaging. See what percents of the packaging are paper, plastic, glass andmetal. Which could be recycled?10) Make a school beautification plan. Figure out how much the paint, plants, flowers,etc., would cost.11) Poll students to see what their favorite outdoor recreations are. Graph the information.How would each kind of recreation affect the environment?12) Is there a land-use committee in your county or state? What do they do?13) Investigate the state parks--how many, how big, number of visitors, problems withlitter, overcrowding or vandalism.14) Investlgatethe cost of ra~s~ng your own vegetables versus buying them at thestore. Consider such things as the cost for rototilling or plowing the soil,fertilizer, seeds, tools, etc. Would you save money by raising a tomato plant ina tub on a patio, balcony or porch? (240

SOURCES OF ENERGY FOR THE U.S.In 1850 In 1970 In 2000(estimated)Source Percent Source Percent Source PercentSun- --Fossil fuel Fossil fuelWood 64 Coal 20 Coal 16Food for work Petroleum 41 Petroleum 37animals 22 Natural gas 33 Natural gas 18Wind, water 7--SunSunFossil fuel Hydroelectricity 4 Hydroelectricity 3Coal 7Misc. Nuclear 26Nuclear, wood 2FROM: Energy Crises in Perspective and Statistical Abstract of the United States 19751) In 1850, what total percent of the energy carne from the sun?2) How does the total percent of the energy from fossil fuel in 1970 compare with thepercent in 2000?3) Does that mean we will use less fossil fuel in 2000 than in 1970?(Hint: Total U.S. energy used in 1970, 67.1 quadrillion units; in 2000, 191.9quadrillion units.)4) What percent of the 1970 total energy used is the 2000 total energy used? (Seeinformation in exercise 3.)5) What environmental problems does each of the sources involve: coal, petroleum,natural gas, hydroelectricity, nuclear?6) Which sources can't be replaced?Permission to use granted by John Wiley and Sons, Inc. 241

Needed:HOW MRNYHORStS IWatch with second hand, metre stick, bathroom scales, flight of stairs1) Find your mass. ___ kg2) On the stairs, measure to see how high the top isfrom the bottom.m\.\EIGI1T3) Walk from the bottom to the top at your normal speed.Record the time in seconds.s4) Calculate your horsepower.Number of horsepower = mass (in kg) x height of stairs (in m)~~~>':::;:"-':':§.!""';~~=.Q_~""::':='-~~='-=="":"'-76 x time (in s)5) Repeat steps 3 and 4, going up the stairs as fast as you can.This gives, your maximum horsepower.Needed:~OW MRNY HORStS ]IWatch with second hand, bathroom scales,gentle slope of known height(1) Find the total mass of you and a bicycle. kg(Lift the bike while you stand on the scale.)2) Record the height of the slopefrom the starting point.m3) From a standing position, ride thebicycle as fast as you can to thetop of the slope.s\.\EIGHT----------------------,~~ ','. ,_. .'~... :.: .':',' ~:.:....... ::: .,:.. ':' ..START.'.',. .: -,'.~", ' .4) Calculate your horsepower with this formula.Number of horsepower = mass (in kg) x height of slope (in m)76 x time (in s)(.242"

269-/ I ----=---::--~::::::=~...l300 -'--320 \ ••\ .. """_ ......._ I "', ' t\(\ f ,'-~ : 377 409 r'.367 \",' ("'\\ \ r---------J___'!. 411 /.406 c.v-;- I 390 '; "-,---_1 ! 46; I • L-_~3_2__ ~, -- ....,_, L___ I \, •, 477 ...... _~ I --- .......,- ... / '(""--f I I , \ ) 3341 I • l I \ ..,436, • I .,.------1 \ f., J 399, --~ 370. ~---.--.,I\, I , • L ~, 398 \484f , 479 "• \ I I \ • '",, 514 I I '466 , /~'\ L I I I '... r(.,J __ -, I -- I. .. / --.\.. ---- _' I I I..r J .t'---/ 399\ ,.... r--__ j' __. '\ ' , 530- - - - -r .L_ - - -4-13- - - - - I _ _ ,-L_.r .. /-'i ' ~._--"'. ,..--- I. 395 376--,AVERAGE DAILYcoY../CfYY\2. FROMI. ISUNI 1 , ~_"\) I I I &480 I • ) __..,.----,- -,I 540 I ,I I 405 -, \ -'If • I I l I J I \ ..."'--- \ I -.... I I \. ...../ I - ........... __." ......... J I \ 429 ...J I "..: \ ...I 549 I • r-----\ I :'I • ' 485 \ 1 \ \......_LI ----- -J 'I 1 \\ , \ r----....- ,.1\ ---~'\, \" 8 439 ,Data from Energy .Primer.....,....~469o !100I 200miles2On the map find the greatest average amount of energy from the sun.The least: callcm 2 cal/cm2. Find a place near you on the map. Suppose you had a perfect 1000surface. How many calories (cal) could you collect a day in that2cm energyplace?3. You need to eat 2200 to 2500 kilocalories of food every day. Would your surfacein #2 collect this much energy?collecting4. A calorie is the amount of heat needed to raise the temperature of 1 gram of water by 1celsius degree. If the calories for one day in 1/2 could all be used; how many celsiusdegrees would the temperature of 1 kilogram of water be raised?5. All the energy used from coal, gas and so forth amounts to that given off by burning7,800,000,000 metric tonnes of coal. This amount is only .0047. of the energy wereceive from the sun. We receive the equivalent of how many metric tonnes of coal insun-energy every year?DATA FROM: Energy Primer, Copyright © 1974 Portola InstitutePermission to use granted by Portola Institute 243

PULL FORPDOLINC~.~(Make plans to take a count of cars travelingcertain routes, at certain hours, onvarious days in your community.Suggestions:Go to the location you have chosen.Count the cars traveling in one directionand the number of passengers in eachcar. Do this for! hour during a morningrush hour, ! hour during an eveningrush hour, and! hour during a middayhou r.Determine how many days you will dothis and vary your days so that someare weekdays and some on weekends.(When you arrive at what you feel is afair sampling determine how many fewercars would have been needed if eachcar would have carried 3 passengers.Determine what per cent of the carscarried only 1 person; 2 persons;3 persons; more than 3 persons.What conclusions can you form as anindividual or as a group carrying outthis project? Can you use these conclusionsto make some recommendationsto your own family (families)? To thestaff of your school? To the membersof your community? To your trafficdepa rtment?(SOURCE:Pollution Problems, Projects and Mathematics Exercises, 6 - 9, Wisconsin Department of PublicInstruction244 Permission to use granted by Wisconsin State Department of Public Instruction

i/' /CE' I~ 0~,~.~ ~'~ f.M .. cOmH~~~~ ~ffYou know that it is more efficient on a trip for a car to have 4 people in it thanjust 1. That way you get more people moved many miles for about the same amount ofgasoline. When two people go 10 miles each, you can say that they have traveled 20people-miles. The number of people-miles per gallon gives one way to compare differentkinds of vehicles.Average number of People in People-milesVehicle miles per gallon vehicle per gallonCar 13.3 1 13.31.3* 17.32 26.65 66.5Bus 5.9 5 29.550 295.0Jet aircraft 0.24 73 17.5Electric train a 1.88 600 1130 (levelground)*Estimated averageaElectricity used is expressed in terms of fuel.1) According to the table, what type of vehicle gives the most people-miles pergallon? The least? _2) Since jets do not give a very highgoing on long trips like to fly?3) Since cars have a low people-milescould take a bus drive instead?people-miles per gallon figure, why do peopleper gallon figure, why do so many people who4) What does the 1. 3 in the "people in vehicle" column mean?5) How was the last column figured out?6) Some cars get 35 miles per gallon. With 2 people, what would the people-miles pergallon be for such a car?7) Suppose a jet had 150 people (including the crew) in it and got the same number ofmiles per gallon as in the table. How many people-miles per gallon would thatgive? Would this be more efficient than 3buses? _8) Besides fuel efficiency, why would one bus with 50 p~ople be better than 10 cars,each with 5 people?_DATA UPDATED FROM: Principles of Environmental Science, by Kenneth Watt,Copyright © 1973 by McGraw·Hili Book CompanyUsed with permission of McGraw~HiIl Book Company 245

(Both the car and the driver contribute to the amount of gasoline used.a)An air conditioner reduces gas mileage*by 9-20%.b) An automatic transmission reduces it by 2-15%.c) Vehicle "eight is the single most important item.A 2200 kg car takes twice as much gas to run as an1100 kg car.d) Emission controls lower fuel economy by 3-18%.e) Unnecessary acceleration can use up to 15% more fuel.f) A tuned car averages up to 10% better gas mileage.g) Cruising at 95 km/h instead of 110 km/h saves 15%,at 80 km/h instead of 110 km/h saves 25%.(Questions1) A driver getting 8 km/f at 110 km/h would be gettingcar were driven at 80 km/h.__________ if the2) Choosing a car with air conditioning and an automatic transmission could lowergas mileage by a maximum of about %.3) A person owns 2 cars--one 2000 kg and the other 1000 kg. If the 1000 kg car isgetting 12 km/f, the 2000 kg car is probably getting onlykm/f.4) Why are emission controls used if they lower fuel economy?2465) *HOW far might a tuned-up car go on the gas it takes to go 1000 km when untuned? (What is a good word for "mileage" in the metric system?Data from Clean Air and Your Car, Environmental Protection Agency

----1) A bicycle can go 18 km/h. How long will it take to go 3 km? minutesA car in traffic averages 21 km/h.3 km? ______ minutesHow long will it take a car in traffic to go2) About 60% of the car trips are less than 4 km. How long would it take a bicycleto go 4 kmat a speed of 18 km/h?3) Almost half of the trips to work are less than 6.5 km. How long would it take tobike 6.5 km at a speed of 18 km/h?4) How far do people in your family go to get to work?How long would it take to bike?5) If bikes were used for short trips to work, the U.S. would save 6% of the 660 billionlitres of petroleum used. How many billion litres of petroleum would thisbe?6) In 1971 there were 24.5% more bicycles sold than cars. There were 11 million carssold that year.How many bicycles were sold?7) One million people get hurt on bicycles every year. What do you think ate some ofthe causes of bicycle accidents?8) Why don't more people ride bicycles? Take a poll of adults in your neighborhood.See how many do ride bicycles to work or to shop. See why the others do not-­distance, weather, hills, safety, too old, afraid bicycle would get stolen.Data from The Bicycle vs. The Energy Crisis, Environmental Protection Agency247

((This includes several ideas from Today's Education, Jan.-Feb., 1975, pp. 90-94.)1. Our energy needs are increasing by how much? (See an almanac.)2. What effects will increased oil prices have on the U.S.?3. Who or what are causing an energy crisis?4. What effects will an energy crisis have on the environment? Will desire forenergy lower air and water pollution standards? Problems with getting fuel:oil spills, strip mining, nuclear plant safety, pipeline digging, emission controls,thermal water pollution,5. What are alternative sources of energy--and their problems?6. Look at ways students can conserve energy at home and at school.7. Write about an imaginary day without oil, gas or electricity.8. Pretend school or family's heat allocation has been reduced 25%. Decide howbest to deal with the problem.9. Make a bulletin board--use graphs, pictures, charts, maps--about fossil fuels, \(sources, uses, pollution and reserves). Find in newspapers or magazines--graphs and articles and advertisements concerning energy. Discuss sources oftheir information and its display.10. Have students list energy-using appliances or objects used in their home. Askeach member of the family to rate the appliances in order of importance to himor her. Display or share the results. Figure the kilowatt-hours used in a year.11. Compare car ads of today and 5-10 years ago. (See old magazines or newspapers inthe public library.) How are they different in what they stress?12. Study the growth in demand for energy in the last 50 years and causes (population,wealth, cars, advertising, development in technology and cheap sources.13. Poll or question adults in the community about causes or cures of the energycrisis. Publish the results.14. In some areas, snowmobiles have become a popular vehicle to own in the past fewyears. How much energy do they consume compared to automobiles or motorcycles?What impact will they have on areas where they are used?IDEA FROM:"Teacher's Guide to the Energy Crisis," Today;s EducationPermission to use granted by J. G. Sartwell, R. P. Abell, and National Education Association(248

C£OJSARYaerosol. A mixture of tiny particlesin a gas. Smoke, fog, mist areaerosols. Many aerosols are storedin pressurized cans (for example,insect repellent).arable.Land that can be farmed.birth rate. Number of births per somany people (or animals); can begiven in percent or in number per1000 people.bronchitis. A disease of the tubesthat go to the lungs.calorie. 1. One calorie is theamount of heat needed to raise thetemperature of I gram of water byone Celsius degree (also calledsmall calorie). Symbol: cal2. (usually spelled with a capital C)1000 Calories. (also called largecalorie) Symbol: kcal; Calcarbon monoxide. A gas produced byincomplete combustion that can't beseen, smelled or tasted. Too muchof it can kill animals that needoxygen.carrying capacity. The number of animalsfor which a region can providefood, air, water, and room.census.A count of the population.compost. A mixture of decayingvegetation. Compost can be addedto soil to enrich the soil.DDT. A kind of chemical that killsinsects. It is usually hard tonotice on food and vegetation.death rate. Number of deaths per somany people (or animals); can be givenin percent or in number per 1000.decibel. A unit for comparing theintensity of sounds. Symbol: dBdensity (population).people (or animals)area.erode.erosion.away.evaporation.to vapors.The number offor each unit ofdoubling time. The time it takes forsomething to double in size or number.ecology. The study of how living thingsare related to each other and to theirenvironments 0emit. . To give off, such as a car emitsair pollution.environment. Evenything that people areexposed to; can also be used for otherliving things (for example, the fish'senvironment).To wash, blow or wear away.Washing, blowing or wearingeutrophication. What happens whenthings get in water and help plantsgrow·, decreasing the amount of oxygenin the water.A way liquids are changedfood web. The relationships amongthings and their food sources.fossil fuel. A fuel formed over longperiods of time from dead animals andplants. Coal, petroleum and naturalgas are fossil fuels.generation. In talking about populations,all the people that have aboutthe same age.geometric mean. A type of average.Technically, the nth root of a productof n numbers.graduated beaker. A container from ascience laboratory that has markingsto show the amount in the container.aer-gra249

ENVIRONMENT GLOSSARYgra-watgram. A unit of mass. A ra1S1n hasa masS of about 1 gram, a tennisball, about 50 grams. Symbol: ghectare. A unit of area equal to10,000 square metres. Symbol: hahydrocarbons. Usually, unburnedpetroleum products in the air. Notharmful in small amounts, but reactwith sunlight to form one kind ofsmog.hydroelectricity. Electricity producedby water power.immigration.or regionoincinerate.Movement into a countryTo burn.insecticide. Any chemical substancethat kills insects.inversion. What happens when a layerof cool air is trapped by a layer ofwarm air above it. Air pollutioncan't escape when an inversion occurs.kilocalorie. 1000 calories.Symbol: kcallitter. 1. Trashscattered around.at one birth (theof 5 puppies).or garbage2. The offspringdog had a litterparticulates. In the air, very smallpieces of matter. They reduce visibilityand contribute to some lungdiseases.pesticides. Any chemical substancethat kills plant or animal pests.pollutant.pollution.Something that causespollution. Whatever makes the environmentunfit or undesirable for living.population. The total number of peoplein a region; can also apply to animalsor plants (the rat population was outof control).produce. (In groceries) fresh fruitsand vegetables.recycle. To use over again after it isth,oloffi away.salvage,smog.2.ingTo save from rubbish.Secchi disk. A black and white diskused to measure how cloudy water is.1. A mixture of smoke and fog.The haze caused by sunlight reactwithair pollutants.soil compaction.soil is.How packed-down the(\(\litre, A cubic decimetre, or 1000cubic centimetres. Symbol: lsolid waste.liquid form.All waste not in a gas ormetric tonne.Symbol: tmicrogram.Symbol:nitrogen oxides. Gases from carsor burning that help form smog.noise.m).l1000 kilograms.One millionth of a gram.Unwanted sound.species. A category in a classificationsystem of living things. Humansall belong to the Homosapiens species.sulfur oxides. A kind of gas in someair pollution. It reacts witn waterin the air to make an acid.thermal.Caused by heat.water cycle. The constant change ofground water to vapor to clouds to rain (/and back to ground water. \250

ENVIRONMENT GLOSSARYsymbolsSymboZsemhakgcentimetrehectarekilogram,e litrem:3mmetTecubicmetreppmtparts per millionmetric tonne251

(Materials are available without oharge unless noted othe~ise.)(SOURCEThe Aluminum Association, Inc.818 Connecticut Avenue, N.W.Washington, D.C. 20006MATERIALSInformation on recycling canscharts.(data andAmerican Can CompanyAmerican LaneGreenwich, CT 06830Information on recycling solid waste.Bicycle Institute of America1923 E. Park StreetArlington Heights, IL 60004Center for Science in Public Interest1779 Church Street, N.W.Washington, D.C. 20036The Conservation Foundation1717 Massachusetts Avenue, N.W.Washington, D.C. 20036Dolphin Enterprises1207 N.E. 103rd StreetSeattle, WA 98125Environmental Protection Agency:Fuel EconomyOffice of Public AffairsU.S. Environmental Protection AgencyWashington, D.C. 20460The Presidenc's Environmental MeritAwards ProgramEnvironmental Protection Agency401 M Street, S.W.Washington, D.C. 20460Public Affairs DirectorEPA Regional OfficeBoston, MA 02203Lifestyle Index ($1.50)A Ci tizen 's Guide to Clean AirECO--An Island Simulation Game ($1.40;upper elementary)Population Eduoation Task Cards ($4.65/set; accessible to middle/junior highschool)Common Environmental Terms: A Glossary(many other publications are for sale.)Miles Per Gallon (test results for automobiles)If one of your classes wishes to get involvedin restoring the environment,you may be able to get certificatesfrom the President recognizing theircontributions. Write for enrollmentmaterials and a brochure containingsome ideas for projects.States Covered:Connecticut, Maine, Massachusetts, NewHampshire, Rhode Island, Vermont(New York, NY 10007 New Jersey, New York, Puerto Rico, Virgin (.Islands252

SOURCESENVIRONMENTPhiladelphia, PA 19106Atlanta, GA 30309Chicago, IL 60606Dallas, TX 75201Kansas City, MO 64108Denver, CO 80203San Frand.s co, CA 94111Seattle, WA 98101United States Environmental ProtectionAgencyOffice of Public AffairsWashington, D.C. 20460Georgia-Pacific Educational LibraryDept. 4900 S.W. Fifth AvenuePortland, OR 97204The Instructor Publications, Inc.P.O. Box 6108Duluth, MN 55806Keep America Beautiful, Inc.99 Park AvenueNew York, NY 10016Delaware, Maryland, Pennsylvania, Virginia,West Virginia, District of ColumbiaAlabama, Florida, Georgia, Kentucky,Mississippi, North Carolina, SouthCarolina, TennesseeIllinois, Indiana, Michigan, Minnesota,Ohio, WisconsinArkansas, Louisiana, New Mexico, Oklahoma,TexasIowa, Kansas, Missouri, NebraskaColorado, Montana, North Dakota, SouthDakota, Utah, WyomingArizona, California, Hawaii, Nevada,American Samoa, Guam, Trust Territoriesof the Pacific, Wake IslandAlaska, Idaho, Oregon, WashingtonFour booklets:Needed: Clean WaterMan and His Endangered WorldNeeded: Clean AirNoise and YouThe Forest and You poster kit (1 kitfree, others 50¢ each)Articles about earth's oxygen supplyEco-Problems posters ($3.50 for a set of12)Ecology posters ($3.50 for a set of 12)71 Things You Can Do to Stop Pollution(write for cost information)National Tuberculosis and RespiratoryDisease AssociationNew York, NYNational Water Commission800 N. Quincy StreetArlington, VA 22203The National Wildlife Federation1412 16th Street, N.W.Washington, D.C.Ranger Rick (10 issues/year, $7.00)National Wildlife (adult reading level,bimonthly, $7.50/year, annual EnvironmentalQuality report)253

SOURCESENVIRONMENTThe Population Council245 Park AvenueNew York, NY 10017The Population Institute100 Maryland Avenue, N.E.Washington, D.C. 20002Population Reference Bureau1755 Massachusetts Avenue, N.W.Washington, D.C. 20036Sierra Club1050 Mills TowerSan Francisco, CA 94104U.S. Department of liealth, Educationand Welfareliealth Resources Administration5600 Fishecs LaneRockville, MD 20852United States Energy Research andDevelopment AdministrationTechnical Information CenterP.O. Box 62Oak Ridge, TN 37830Studies in family planningReports On populationCountry ProfilesInterchange (teacher's newsletter)World Population Data Sheet (35¢)Population Report (monthly newsletter)Population education teacher packetsInformation on speakers and filmsMonthly Vital Statistics ReportWall Chart: Energy History of the U.S.,1776-1976Many pamphlets and booklets: EnergyStorage; Geo-thermal Energy; EnergyTechno logy; . . .One can request information on a specificsubject (e.g., solar energy).(('(254

;to MATHEMATICS ANV MUSICIt is said that music is a universallanguage--understood and enjoyed byall. This is certainly true for mostmiddle school students. Even those whodo not participate in an orchestra orband will dance, sing or keep theirears glued to a transistor radio or setof headphones. This interest in musiccan provide the motivation for individualactivities or for an entire unit inmathematics and music.Where does the mathematics come in? Today students learn music and learn mathematicsas different courses without associating music with mathematics, but the twosubjects were not always separated. For centuries the Greeks viewed mathematics andmusic as part of the same subject. Modern students can repeat some of the experimentsof the Greeks by investigating the ratios of the lengths of strings and of the tonesthese vibrating strings produce. They can also study the graphs of the pitch, loudnessand quality of tones.The activities in this collection are organized into four sets. The first setrequires no knowledge of musical notation or expertise in playing an instrument. Inthis first set, students can research the financial aspects of a musical concert;determine the meaning of the different speed settings on a record player; determinethe pressure exerted by a phonograph needle; use logical reasoning to classify instruments;and solve mathematical word problems about music and musicians.The next set of activities (beginning with Notes, Rests and Fractions) involveslearning and practicing musical notation. The activities stress the relations betweenfractions and notes, rest, and time signatures. Students are asked to use the ratioidea--if a quarter note is assigned one beat, a half note is assigned two beats--ina variety of activities. Most of the activities in this set can be used with studentsprovided that the musical notation is explained. Musically inclined studentscan be an information source for the rest of the class. The Britannica JuniorEncyclopedia has an easy-to-read article on music notation, and the school musicteacher could also provide some background. The glossary at the end of thissection will help in many cases. You will probably want to work through the activitiesyourself to be sure they are appropriate for your class.255

INTRODUCTION MATHEMATICS AND MUSICMotions and Musio is the first page of the third set. It describes how aseries of notes can be reflected (played backwards), translated (repeated or slidup or down on the scale~ and rotated (turned about a point) to compose a piece ofmusic. Students who do not read music can do the activities by applying the motionsof geometry to the notes. Someone else in the class· can play the compositions Onthe piano. Other activities in the set also involve analyzing and composing music.Some of these are quite simple; others would make good projects for an individualstudent or for a group of students who are very interested in music.The fourth set concerns the subject of sound as related to music. Shapes ofSounds describes graphs for sounds. Students are asked to compare the loudness andpitch of two sounds based on their graphs. (An oscilloscope would make this activitymore meaningful, but it is not necessary.) Some activities in this set usegraphs or charts giving the range in pitch or loudness of voices and instruments.Students are asked to use these charts and graphs to compare the instruments. Otheractivities ask students to order sounds according to their loudness, discover relationshipsbetween frequencies of notes, and investigate various sound phenomena. TheBritannica Junior Encyclopedia has an excellent overview of sound.Mathematics has played an important part in modern music. It has been used toanalyze music; to help create buildings with good acoustics; and to develop radios,phonographs and amplification systems which reproduce music as accurately as possible.On the other hand, there are many people who have seen music in mathematics.Pythagoras thought music followed the rules of whole numbers, and Kepler looked formusic in the spheres. The page of quotes following this introduction gives someadditional thoughts of mathematicians and musicians about the relationships betweenmathematics and music. Though not all of the possibilities in mathematics andmusic are presented here, perhaps these activities will provide new, interestingand useful information for teachers and .students alike.(\((256

INTRODUCTIONMATHEMATICS AND MUSICMathematics is music for the mind; music is mathematics for the soul..--AnonymousMusic is the, pleasure, that the human soul experiences from counting without becingaware that it is counting.--LeibnizMathematics and music! The'most glaring possible opposites of human thought!Yet connected, mutuaUy sustained!--Hermann von HelmholtzThe musician feels Mathematics, the mathe­Mathematics is the music of Reason.matician thinks Music.Mus{c is number made aud~ole,--J. J. Sylvesterarchitecture is number made Visible.--Claude BragdonThe science of Pure Mathematics, in its modern developments, may alaim to be themost original creation of the human spirit. Another claimant for the position ismusic.--A. N. WhiteheadSuacess in musia and mathematics also depend upon very much the same things--finetechniaal equipment, unerring precision, an abundant imagination, a keen sense ofvalues, and, above aU, a love for truth and beauty.--Elmer B. ModeFor three bars I Zisten to it; thereafter I distinguish nothing, but give myselfup to my thoughts. In this way I have solved many a difficult problem.--LagrangeMathematics is the inspired ordering of an infinite world of numbers. That'sOswald Spengler's definition. Music is the inspired ordering of an infinite worldof sound. That's my variation on Spengler's theme.--Howard p. LyonIt is the same with music.I'm glad he did . •• MusicalYou have to make them practice. My father did, andtraining helps their arithmetic.--Arthur Fiedler257

Listen to the Students~ TO STUOCt\lT INTERESTS:Let the students start the processfor you. All you need to do is listento their conversations and questions.You probably already know that some oftheir discussions involve recordingstars, records or albums and, perhaps,concerts. If they playa musicalinstrument, that is also a specialinterest.Ask them questions about theinstruments they play, the concertsthey attend, or musicians or composers they like. If a concert is being held in thearea, there might be a discussion about it. Students might talk about plans to go,or that they would like to go, and afterwards, about the performance. When you hearthese types of conversations in your classroom, take advantage of it. An activitylike In Concert can be used at this time. (Use the Students as a ResourceStudent background in mathematicsand music can be helpful to a teacherwanting to emphasize relationships betweenthe two disciplines. Knowledgeof their experiences will help determinewhether some activities are indeedappropriate and can also lead tonew ideas for material.YOU TWO PRoBABL'I KtJOW MOR.E.ABOUT MUSIC THAt-! I 00 - WOULDYOU BE: WILLlIJG TO HELP OTHERS Ir-JTHE CLAS,S?This information could be filed for future reference by both you and your students.Here are some questions you might want to ask.1. What kind of music do you like to listen to or play?2. How often do you listen to the radio? records or tapes?3. What instrument do you play?4. In what group(s) have you sung?5. Are you taking music lessons? music classes?(258

particularlyhelpful during individual and group protimeby con-jects. Students may savesuIting the file for a specific resourceperson instead of asking around theclassroom and bothering other workinggroups. Students are also answering forthemselves an often asked question:"Where can I find out?" Pupils gain fromthe experience by utilizing a problemsolvingapproach and from the sharing ofinformation--a good start toward helpingstudents learn from each other.I FIGURE:O OUT HOW TOWRITE. SOME NO,ES, BU, J:~"\LL CAN'T PLAY '\lEM. 1.c;.UES'1> I !VEEP A. PIAIIlOPLAYER.Read Newspapers and Magazinesmusic.Start by reading your own magazines and newspapers and saving items related toStudent sheets can easily be made with these articles and advertisements.BLANKCASSETTE TAPE60 Min. Play 90 MilT. PlayAeg. 1.49 Reg. 1.99Which tape has the higher moneydiscount?What is the percent of discountfor each tape?Which sale price is the betterbargain?An example is shown to the left.mances can be used for ratio and proportion,percent, money and consumer problems.Statistical graphs can be usedfor interpretation questions, or studentscan gather information and display it ina graph.Articles and advertising make goodtransparencies for use on the overheadprojector.While the information is onthe screen, the teacher can ask relatedquestions.Clippings can also be displayedon the bulletin board.Theteacher can make up questions related tothe article or the students can writethem.Student questions can then becollected, sorted and used on a laterlesson sheet.Advertising in newspapersand catalogs for equipment or perfor-259

(PAGE '2»(Ask students to look for informationfor you, perhaps as a special project.Their magazines at home areprobably different from yours and willincrease your source of materials.LOOK Al nus ARTICLEI rOUt-JD.Survey your school library for information in books and magazines.Time, Newsweek,Saturday Review and Popular Science each have separate music sections in eachissue.sources.Scientific American, Consumer Reports and Rolling Stone are also goodThe librarian, music teacher and local disc jockeys could have good, valuablematerial.An annotated bibliography that lists printed and audio visual sources is providedat the end of this section.260Try Some ProjectsStudents often enjoy working on projects, particularly with a topic they alreadyfind interesting.Their conversations about musical instruments, current hits andfavorite performances can be adapted to individual or group projects. (If you or your students have had little experience with projects, start withsimple, short-term ideas.help.Some examples are:1) Find articles or advertisements with information related to music. Make upsome mathematics questions and turn them in with the article.2) Conduct a poll at school togather responses to a questionof interest (examplesare on a student page TheMusic Market).The first few projects you assign will need the most structure and teacherStudents will have to learn how to use a problem-solving approach, to gathertheir statistics, to compile them and to convey this information.projects.After some common experiences with simple problems, gradually try longer-termThe student sheet The Music Market is designed for ideas on this level.Regardless of the length or nature of a project, students will have questionsconcerning procedure.Providing an example or making a poster that outlines thesteps for a particular project are very helpful to students. As projects get more (involved the students can work more independently.

MATHEMATICS ANDMUSICTITLEPAGETOPICMATHTYPEIn Concert264Researching musicaleventsInvestigating financialaspects OT public eventsWorksheetActivity cardAs the Record Turns266Finding a meaning for thespeed settings on recordplayersUsing rates and ratiosActivity cardThe Pressure's On268Understanding why a heavyphonograph arm can harmrecordsComputing pressureWorksheetI Believe in Music269Experimenting with thetempos of music using ametronomeUsing rates and ratiosActivity cardPuzzling Prefixes270Determining words relatedto musicUsing numeraical prefixesRecognizing names ofgeometric figuresWorksheetPuzzlEi'Musical Time Lines271Studying the life spansof composers andmusical stylesUsing number linesTeacher ideasBulletin boardThe Music Market272Examining cost of musicalinstruments and equipmentDetermining people's preferencesin musicCollecting dataComputing costs and bestbuysOpen-endedprojectsInstruments withClass 274Comparing and classifyingmusical instrumentsClassifying objectsThinking logicallyTeacher-directedactivityAttribute Games 279Comparing and classifyinginstrumentsPlaying logic gamesGameNotes, Rests, andFractions 283Learning the symbols andtime values for notesand restsUsing fractionsUsing ratio ideasTeacher pageBicentennial Beat 285Determining appropriatetime values for notesUsing fraction conceptsUsing ratio ideasWorksheetTransparencyAnd the Beat Goes On 287Determining beats in ameasureChecking accuracy of timevalues in written musicUsing fraction conceptsUsing ratio ideasWorksheetTransparency261

(TITLEPAGETOPICMATHTYPEYou'd Better CountOn It288Examining baton patternsfor music conduc"torsCounting measures of notesand restsCountingWorksheetAc"tivity cardA Symbolic Story289Recognizing musicalsymbolsRecognizing mathematicalsymbolsStoryPuzzleMatiens and Music29lSeeing patterns in musicComposing musicApplying the reflections,translations, and rotationsof geometryTeacher pageReflections292Seeing reflectionsin musicApplying reflectionsTeacher pageFirst Movement:Reflections293Finding reflections inmusicReflecting a series ofnotesApplying reflectionsWorksheetTranslations294Seeing repetitionsin musicApplying translationsTeacher page(Second Movement:Translations295Repeating phrases inmusicIntroducing transpositionApplying translationsRotations296Applying ro·tationsTeacher pageThird Movement:Rotations297Writing notesApplying rotationsWorksheetVariations298Composing musicCombining motions ofgeometryTeacher pageFinale:Variations299Composing musicCombining motions ofgeometryWorksheetSimple MusicalInstruments300Making simple musicalinstrumentsUsing volume and ratiorelationshipsActivity cardIt Has BeenFrequently Noted30lDetermining the key ofa piece of musicGathering dataMaking tables and graphsActivity cardBeethoven Bargraphed302Analyzing music todetermine the composerCollecting dataMaking and interpretingbar graphsTeacher idea(262

TITLEPAGETOPICMATHTYPEMod Composition 305Composing musicUsing number sequences andmodular systemsTeacher ideaA Chancy Composition 306Composing music andevaluating the resultingcompositionProducing random results byrolling diceActivity cardAn OrderlyArrangement 307Realizing that there aremany possible melodiesDetermining the number ofpermutationsWorksheetThe Twelve-ToneMethod 308Composing music with thetwelve-tone techniqueUsing geometric motionsApplying permutationsTeacher pageShapes of Sound 311Examining pitch, loudness,and quality of soundgraphicallyReading graphs of soundsWorksheetsActivity cardsCan You Hit High C? 313Comparing the range inpitch of voices andinstrumentsReading a graphWorkshee"tActivity cardAre We on the SameFrequency? 314Studying frequencyrelationships in tonesof differing pitchDiscovering numberrelationshipsActivi"ty cardWorksheetNumbers for Sounds 315Understanding soundComparing loudness ofsoundsOrderingComparing measurmentsActivity cardWorksheetThe Sound of Music 319Learning symbols forlevels of loudness inmusicComputing differencesWorksheetActivity cardUsing a DecibelReader320Comparing loudness ofdifferent types ofinstruments and musicReading a guageComparing readings ona guageOpen-endedprojectsSound Projects322Exploring soundOpen-endedactivities263

(----:--0-=---- APPEAREDWHOIN =-------CI"TVAT THE ------."""ur.;;:----­BUILOlW,

UseThe general questions asked on the student page are one way that a teacher couldbring information about musical performances into the mathematics classroom. The pageis, of course, most appropriate when students have recently attended a concert. You maywant to add specific questions to the page for a particular concert.Some questions require access to specific information. Newspaper articles, postersand the promoter are good sources. If the information is not easily obtained, then rewordor omit those questions.If concerts are not held nearby so that students can attend, discuss one that theywould be interested in attending. A student page could be developed using an advertisementand an article with information about a performance.The questions could be adjusted to ask about a television special in which a favoritesinger hosts or performs. Athletic events or plays are topics that can be similarlybrought into the classroom. Writing the questions could be a student's project. Schoolproductions provide much interest and data. Information about attendance capacities,ticket sales and expenses is readily available and of concern to many of the students.Extensions of the Student Page:Some students may finish the questions feeling that the promoters make a lot ofmoney. If this interests them, they may want to investigate the business of promotion.Suggest that they write or call agencies to find information. A common question is: Howmuch do various performers earn? (The Guinness Book of World Records has interestingfacts about musical performances.)Project Idea:Plan a money-raising project that brings in a star performer.be involved in determining the expenses and estimating the possible profits.Much computation willSuggestion:Keep the data about musicians, fees and performances.of study such as statistics.It can be used in other units265

(Materials Needed:Record player with variable speedsISeveral popular 333 albums, 45 singles and a 78 record (if available)Stopwatch or clock with a second handQuestions:How fast does a record turn?What do the speeds on a record player mean?1.IPlace a small marker on the outer edge of a 333 record album. SetIthe record player to 333 and carefully count the number of revolutionsthe marker makes in I minute.below and record the number.accuracy.Make a table like the oneRepeat the count two more times for(minute Iminute. 2.,Mlnu.te 3Nu.mbe.... Or"evolutionsFind the sum of the revolutions.Find the average by dividing the sum by 3.II. Repeat the activity again using a 45 record. Make atable.III. If you have a 78 record, repeat the activity again.IV.Place a small marker on the label ofIa 333 album.Repeat the activity. Find theaverage for the marker on the label.found in part I?How does this average compare to the average(266

The following are additional activities and questions that can be used asa follow-up to the As the Record Turns student page.(a)(b)1Play a 33) album at 45 rpm. How is the sound distorted? Howmuch faster is the recora revolving compared to its normal ormalspeed?1Play a45 single at 33 rpm.3How is the sound distorted? Howmuch slower is the record revolving compared to its normallormalspeed?Note: Most record players have a different needle setting for 78 rpmrecords, so you should not try to play one of these records at a differentspeed.(c) Measure the time a song plays at its normal speed. (This figurecan also be found on the label.)song played at a slower or faster speed.Compute the time of the sameCd) If a song takes 3:,30 minutes (3.5 minutes) to play at 45 rpm,(e)(f)how many revolutions does the record make?If it takes 21 minutes to play one sidehow many revolutions does the turntableof an album at 3~ rpm,3make?How much farther does the needle go in one revolution when it isat the outer edge of the record than it does at the inner edge?Why doesn't the pitch of the music change as the needle getscloser to the center of the, record?Did you know that • • •the first needle used to play records was a cactus needle?78 rpm was the first speed used for records because it seemed likeaconvenient speed?RCA tried to corner the market on records when they patented the45 rpm single record with the large center hole?with the invention of more refined and sharper needles, records1could be made with finer grooves which played best at 33 3 rpm?some commercials played on radio stations run at 16 rpm and startfrom the center and play to the outside? 267

PRE.5'2>URE: I~ GIVE./-.l l"l 'ORee: (POUtJDS)PER SQUARe:: UI\.)IT.WHICHEXERT5 MORE. PRESSURE:?It makes sense that anelephant exerts a lot ofpressure on the ground, butwhat about the needle of aphonograph?THE £,U:'PHAl\lT OR THE PHOl\lOGRA,PH t-JE:E:OU::'?1) The oval point of a phonograph needle fits into a .0002-inch by .0007­inch rectangle. The force of the needle on the record is .0033 pounds.(If .0033 pounds of force would be on every .0002-inch by .0007-inchrectangle, how many pounds of force would be on one square inch?2) A large elephant weighs about 10,000 pounds. Suppose an elephantthat weighs 10,000 pounds stood on one of his legs. Suppose the foothe is standing on covers one square foot. How many pounds of forcewould be on one square inch?3) Which exexts more pressure (force per square inch), an elephant or aphonograph needle?4) The information given in problem 1 was for a high quality needle(stylus) and turntable. Do you see why some heavy arms on phonographscould ruin records after several plays? (268

Materials Needed:Dtt=b¥fieve jV) mu.si C I'4/Records and record player, clock with second hand,metronome, piano, drums, guitar, flute or otherinstruments, sheet music.I. (a) Select several records so you have musical pieces withdifferent tempos (beats per minute), for example, aslow country western song, a Sousa march, a rock androll piece, and a classical arrangement.(b)(c)(d)(e)(f)Determine the tempo of the song by counting the numberof beats in 10 seconds. You count the beats by tappingor setting a metronome.Repeat the count to check for aemuracy and record theresults in the table as a rate: number of beats 10seconds.Rewrite the rate and express it in the table as number of beats : 60 secondsLook at the record to find the total time of the song and record the time inthe table.Estimate and record the total number of beats in the song.Musica.l Number of' Numbe.r of Tota.l time. Estimate of totQ.1Selection bea.+s: 10see. bea.1s: GOsec. of ~on('] nlJ..,bevot bea.tsJ,. : IOsee. : GO§e(.2. : :3- : :4 : :5 : :G. : :II. Have a classmate or the music teacher playa selection at various tempos. Use ametronome to set the tempo and keep it even.A classmate could keep the beat on a drum or by tapping a pencil. Time a song ata specified tempo and record the time in the table. Select a new tempo and estimatethe new time for the song. Check the estimate by having the musician playthe song at the new tempo.Musical Tempo ::: Toto.. \ til'TlE. New tempo" Est'll'TlG-tedSelection bea.+s: Iminu.te of sona bea.ts : Imi"u+e time of SonqJI.: :2. : .3. : :4. : :III. Select some sheet music. Read the tempoestimate the tempo by tapping a pencil.Have the musician play the selection.suggested on the music. Have a classmateCheck the estimate with the metronome.J269

Many musical terms have prefixes which have a numerical meaning.and a septet is a piece of music for seven performers.an eight-note chord.with this crossword puzzle.,.."a. Hexagonb. Trisectc. Octagond. Hexahedrone. Quartf. Octahedrong. Quadrilateralh. Triangle270Septa means sevenOcta means eight and an octad isKnowing the prefixes which have numerical meanings will helpA dictionary might also be necessary.2.4.6.~~ 5 7.9.AcrossMusic for two performers.Music for four performers.A group of three notes ofequal value to be performedin the place of twonotes of the same value.Music for three performers.A chord consisting of threenotes: the root note, athird and a fifth.11. An interval consisting of 8tones from a major scale(12 half tones). An intervalfrom one C to the next (higher C, from one G to the ,next higher G, etc.12. A single unvaried tone; Aperson who stays on onenote while singing.Downl. Music for six performers.3. Music for five performers.5. A six-note chord.8. Music for eight performers.10. Music for one performer.The mathematical terms listed below have prefixes similar to those in the musicalterms in the crossword puzzle.Write the letter of the term next to the correct meaning.A unit of liquid measure.A polyhedron with six faces.A polygon with three sides.A polygon with four sides.Four of these make one gallon.To divide into three parts of equal measure.A polyhedron with eight faces.____ A polygon with six sides.A polygon with eight sides.((

Time lines can help students seG which composers lived at the same time and whichcomposers could have influenced later composers. Students could look up the life spanof their favorite composers and place them on a time line. A large time line could bemade on a bulletin board for discussions. A short list of composers is given below witha time line made from the list.J.S. Bach 1685-1750Handel 1685-1759Haydn 1732-1809Mozart 1756-1791Beethoven 1770-1827Chopin 1810-1849Liszt 1811-1886Tchaikovsky 1862-1918Schoenberg 1874-1951Gershwin 1898-1937Vaughn Williams 1872-1958Copland 1900-Barber 1910-I-M07.ART-lI-VA\JGHtJ WI LLIAIV\S-ll--I-IAYPE.~---iIf---L1""7.T--;I-

(Select aproject onthese or your own idea.which youThink ofwould like to work.It can be one ofways to share the information you find.1) Where is the best place to bUy recordalbums?to bUy tapes?•••••••••••••• •••••••• •••• •••• . :.. :::. r-::;:.: /. ", """):kJIMPLb:~ kJb:COkJD kJ~C~ ° ,~\s~~~~:f ~IE°h:,owoC!oC!oC!o:U:oi:oi:i:i:i:moU:;wC!o:u::n:i:i:i:i:iJoU:;o;:;ocuo:u::U:oui:ommqowC!oC!o:n:o"- • REC.ORD ~ l'APECb 0).1 SAL£.!LATE:5T HIT5 43.99 l

4)THE ~nlJ§I[~RIKEi(CO"lTI~UW )What are the price ranges ofindividual instruments?do the new and used pricescompare on instruments of thesame quality?Are there anyinstruments that increase invalue as they get older?I REI\lTE:O A VIOL/\FOR MY ORGIESTRACLASS.HowI LEASE.D A GUITP-R.I"OR PRIVATELES'50/IlS. r-__ARl;; 'IOU R~ALLVWORTH 400 TIMESMORE: MO~EYTHP-M I AM?5)Compare the cost of purchasinga new instrument,with renting monthly orleasing with an option tobuy.Which is the bestarrangement and under whatcircumstances?6) Compare the prices of differentbrands of radios, tape recordersor stereo components. What newfeatures are offered at eachjump in price?~ -::=::....... . ., .;;~~~l&~~,i7) What percent of your classmatesplay a musical instrument?Have their own stereoset? Play in the schoolband or orchestra?8) Take a survey to determine theschool's or class's favoritesinger, recording group orkind of music.273

INSTQUM~NTSWIT~ CLRSSClassification is important in mathematics,biology, chemistry, music and many otherfields of study.In mathematics numbers areclassified as whole numbers, rational numbers,irrational numbers,.•• and geometric figuresare classified as polygons, polyhedra, curves,squares, '"Biologists classify plants andanimals and chemists classify the elements andMATHEMATICALLY,'IOU l-ll\V£ MORECLA~'" 'l-lA\Il I DO,compound substances. Music and instruments ,-----,each have their classifications. Althoughclassification is used in many subject areas~,----;===~~~==JC::~the study of classes, often called sets, is abranch of mathematics and logic.(BUT '/OU PLAYSUc.t\ Bt:AUTIF'ULMUSIC,RECTA\IlGLEPARALLELOGRAMQUADRJLAT£R.ALAttributes such as number, size, color or shape can be used to classify objects.Students can decide how to classify a large group of objects, determine relationshipsbetween the sets (the class of whole numbers is a subset of the class of rational numbers)and operate on the sets (what numbers belong to both the set of prime numbers andthe set of odd numbers?).(Focussing on properties or attributes allows for more flexible or creative problemsolving. If an object is unfamiliar to someone, its name alone does not help describethe object. A list of its properties will help someone visualize the object. Also,if you need an object that is not available, a list of its properties could suggest asuitable replacement.(Trombones can play some cello parts and have been used as "substitutes"in small orchestras.)A collection of musical instruments can become that large group that needs to beclassified by students.Most people tend to think about only the names of the instruments,but their use is really a function of their many attributes.The activity onthe next page suggests that students explore the classification of instruments and howthese classes relate to each other.Further Sources: Learning Logic, Logical Games, Vol. I (First Years in Mathematics)by Z.P. Dienes and E.W. Golding, New York: Herder and Heider, 1971Describingmatics andversity of& Classifying, Unit 3 (Minnemast) by Minnesota Mathe-Science Teaching Project, Minneapolis, Minnesota: Uni­Minnesota, 1967('274

INsTQUMt;NTSWIT~(PAGe 2.)[LRSSHave the students give the names of familiar musical instruments while one studentlists them on the board. Other instruments can be added to the list as students thinkof them. Have band or orchestra students bring their instruments so the instrumentscan be compared. The pictures on the following pages could be used as a substitute forthe actual instruments. Pick two of the instruments and have students describe how theyare alike and different. You might want to list these similarities and differences onthe board. For example, a trumpet and a trombone both start with "tr", are (probably)gold colored and are called "brass instruments." A trumpet is smaller than a tromboneand a trumpet can play higher notes than a trombone. A violin and a trumpet are bothsmall, but one is a stringed instrument while the other is brass.Discussions continue until students realize that certain instruments--based on theirattributes--can or cannot be members of select groups. For example, a trombone cannotbe in a woodwind trio, but it can be in a brass quartet. If recognition of the instrumentsis a problem, pictures could be put on the bulletin board, along with the threemain classification lists. Cassette recordings of each instrument could also be playedto he'·, familiarize students with their sounds. See if they can recognize 2 differentinstruments playing at the same time.At some point suggest that the students decide on a listing of several ways to classifyall instruments. Common but simple classification could be this:SIZE:SOUND:TYPE:large, medium, smalltreble, bass, full range, non-tonalwoodwind, brass, string, percussion,electronicSize will be the most commonly argued. A trombone may seem large or medium sized,depending on what other instrument they are compared to. "Sound" is usually agreed asthe pitch of the majority of its notes, not the clef from which the students read music.Ask for methods to arrive at a standard by which size can be judged. One suggestion isto start with a listing that they do agree on:small---flutepiccolomediumsaxophonelargebass drumpianotubaorgan275

INST~UMtNTSWIT~CLRSS(PAGE 3) (After listing the large instruments, students will find it easier to add to the listof medium ones. Discuss how determining the two outer limits (having examples of largeand small) helped set the middle limit. Use that list to see that: (usually)small instruments are held up,medium instruments are held down or to the side,large instruments rest on the floor or a standand the player sits next to it.Some common results of the attribute lists:SIZE:SmallpiccolofluteclarinetviolatambourineviolintriangletrumpetMediumcymbalssnare drumguitarbanjotrombonesaxophonebaritone hornfrench hornLargecellobass (double)organpianobass drumtubaharptimpanisynthesizer (SOUND:TrebleclarinetflutepiccolosaxophonetrumpetFrench hornviolaviolinbanjoBasstrombonebaritone horntubatimpanicellobass (double)Full RangepianoorgansynthesizerguitarharpNon-pitchedsnare drumcymbalstriangletambourineTYPE:WoodwindclarinetflutesaxophonepiccoloBrasstrumpettrombonebaritone horntubaFrench hornPercussioncymbalstrianglesnare drumbass drumtimpanitambourineStringguitarviolaviolincellobass (double)banjopianoharpElectronicorgan (most)synthesizerAsk students if they see relationships between some of these attributes.probably notice that small instruments are often treble, that the large onesbass or both ranges, and that brass are either medium or large.276They willare usuall~

INSTRUMtNT~WIT~(PAQ£ 4)[LRSSPIANOSYlJTI-!€5IZERORGANTAtllBOURINEeo",s.. DRUMSAXOPHOIJEVIOLINTR\A\\lGLt:eo"'RITOIJE:HORl\J277

INSTRUMtNTSWIT~ [LRSS(PAGE: 5)(GUITARCELLOTROMBOtJESfJAREDRUMCLARlI.1£T(TRUMPETL..- --L_::::::.. .....L__~~..::l!~_BANJODOUBLEBASS____I(278

I. Teacher Introduction to Game: Use attribute lists compiled in theactivity Instrwnents with Class. Ask students to name the instruments,given any three attributes.Examples:Possible Answers:a) large - bass - brasstubab) small - treble - woodwind clarinet, flutec) medium - bass - brasstromboned) large - non-tonal - percussion bass drumII. Players Needed: I to 436 attribute cards--each withone word. Use these words:small, medium, large, bass,treble, full range, nontonal,brass, string, woodwind,percussion, electronic.Make 5 cards for each word.Stack cards by attributeface down in center.000instrumentsize sound type:tILPlaycards (see previouspages) spread face up in viewof all playersFirst player draws a card from each attribute pile. He then selectsan instrument card that fits the attributes. The other players agreeor disagree and the following points are given:3 points: instrument does have all three attributes2 points: instrument has two of the attributesI points: instrument has only one of the attributes0 points: instrument has none of the attributesPoints are recorded.on the bottom of eachshuffling them.First player puts back attribute cards eitherstack or in each stack somewhere, slightly re-Play continues as each player draws atributes and selects the appropriateinstrument. If there is no instrument to fit a certaindescription, player can indicate that and still receive the 3 points.IV.WinnerThe winner can be the first person to reach an agreed on score or theperson with the highest score after a given time limit.279

(A GAME IN RECOGNIZING ATTRIBUTES OF INSTRUMENTSMaterials Needed:Instrument cardsObject: Make a path with the instrument cards so that adjacent instrumentsvary by only ~ attribute.Play:Divide instrument cards among players.Player I picks one of his instrumentcards and places it in the center.0.--...-.,..........,,0 ME:DIUM~TR.I::eL. E.e.RI'-5~Player 2 places one of the instrumentcards adjacent to the first card.This instrument can vary by only oneattribute. ---o'"..'"MI::DIUf""Player 3 can place aend of the path.card at eitherPlay continues until a person (winner)gets rid of all his cards or nobodycan make a play.A player gives up a turn if one of his instruments can't be addedto one of the ends. A player who places an instrument card incorrectlymust take the card back and give up that turn.(280

Playing board withInstrument cardsAttributes listsquare gridAll instrument cards arethe center of the table,playing board.face up inaround thePlay:Players take turns placing instruments in squares on the playingboard. Any card which is placed so it touches another card musthave the following attribute relations to the original card.----Points are given as follows: c> 0 Io

(Materials:Playing board with 2 large intersecting circlesSet of instrument cardsAttribute cardsAttribute listObject:To guess what attribute the circles representPlay: Leader selects 2 attributes from the cards and imagines the 2circles as the corresponding attributes.Each of the players in turn picks up an instrument card and handsit to the leader who places it in the appropriate area on the playingboard. There are 4 areas that an instrument can be placed: inthe space shared by both circles, in the part of each circle notsILared by tILe other, or outside the circles.Example of play if A is "large"and B is "$tring"Player 1 selects guitar soit is placed within "string"Player 2 selects tuba, andit's placed in "large"(Player 3 picks clarinet,which is neither attribute,so it's placed outside.Player 4 picks cello withboth attributes, so it'splaced in their intersection.Play continues until there are enough instruments in place fOr oneof the players to guess at the two attributes. One can guess onlyafter his instrument card has been put down and he must give bothcircles correctly. If the guess is correct, he receives as manypoints as there are instrument cards left on the table. If theguess is not correct, he subtracts 5 points from his score. A timelimit or point limit can end the play. The winner after that limitcan be the next leader.282Extensions:1. Add more instrument cards2. Add another circle (andtherefore another attribute)3. Add negation of attributesto deck of attribute cards,ex: not brass, not large(

NOT£S,~£STS~~~ r~f1CrIOrYSFractions play a vital part in music.Music cannot be composed and musical scorescannot be read without some understanding of the fractional values represented bymusical notes and key signatures.introduction to musical notation will be necessary.Before using the following student pages, a briefSome suggestions are given below.I. MUSICAL NOTES AND RESTS ARE GIVEN FRACTIONAL NAMES.A whole note may be played for different lengths of time in different pieces of1music. If it is played for 1 second, then a quarter note would be played for 4second. If a whole note is played for 2 seconds, then a quarter note would beplayed for 21second and so forth. The same relationship holds for rests. (In fact,the rest remains to be seen and not heard.)NOTE NAME REST NAME0 whole noted half note-------halfwhole restrestJ quarter note I quarter restJ eighth note '1 eighth restJ sixteenth note :; sixteenth restII. TIME SIGNATURES LOOK LIKE ACTUAL FRACTIONS.The top number refers to the number of beats in each measure.designates the kind of note that receives one beat.The bottom numberI") I I I I/ .....J A lIIIIIt 1- J I 1\ I 1\ I(( "\ / l- T -' - C :1 J... J ..\: / , It-TIME 5lGt-lATURE- 3 BEATS PE.R MEASURE- QUARiE.R NOTe GE,S O\\lE BEAT,JvOIJE. MEASURE -3 BEAlS,• BARSSEPARATE. Tfi£MEASURE.S283

284NOTtS,~£STS'-tiD r~l1CrrONS(COIJTI "lUE:D) (III. EXAMPLES FROM ACTUAL MUSICAL SCORES ARE GIVEN BELOW.NOTESIn the score to the right, aquarter note is given one beat.A dot adds half the value ofthe note.od.cdoJ.oJJwholedotted half ----,.... 3 beatshalfBEATS----•• 4 beats-----•• 2 beats1dotted quarter ~ 1 2 beatsquarter------I... 1 beat3dotted eighth -""4 beatIn the score to the right, ahalf note is given one beat.NOTESBEATSo· dotted whole ~3 beats0cd.wholedotted half.. 2 beats1-..1 2 beatshalfcd.1 beatdotted quarter-.-3beat.J. 4quarter .. 1oJ 2 beatSuggestion:A "pick up" note.~~ +}4J~This note plus the "pick~ no,. "". 4 ' ••".,~~J eighth1------.. 2 beatJ dotted sixteenth-. ~ beatJ sixteenth ------l.... 1 beatij~~oP=~~~~,~'-../$oemr~JJ eighthdotted eighth ~~ beat41----... 4 beatGive students a few bars from a musical score and have them complete achart like those given above.up1t((

grCtNTtNNIRL8tRTOne or two notes are missing in each measure.note of the correct time value.Replace each X with aCan you name this familiar song?"pick up" note\ , tD L.. ( I I I I,/ 1..., v t+ I I l I I , I I X I{( ,v/') IA I • ,e· I'• Ie· I I ,e x•"- !J v~ e x -I e-t\ ,IJD k/ I •\. j vII ,•. Ii Ie. x x I) I- -_- . - -I-x --r-I I-,iT D I... r/ IA'V II I r- 1- _ X r I x III YDL.. I I h I Iej,I~ "~x" ,!J I? I.- -. I ,-This note plus the "pickup" note gives 4 beats.-t) L..,V ~ " ~ I I .. r- I -(( Y D l.- I r r I x CIIIP~ I\. ) II I) I-r285

I~::::::::========-- ('(286

RID THE BERTCO[5 ONwhole

lbTTbU[CJUf\JT fJI\j IT! (Counting is basic to number and mathematics, but it is also veryimportant in music. Not only do musicians have to count out the numberof beats per measure, they also have to be able to count a large numberof measures of rests to be sure they rejoin the band or orchestra atthe correct moment. It can be very embarrassing to count incorrectlyduring a concert and come in at the wrong time.I. Here are some common conducting patterns. Each pattern representsthe beats in a measure. Determine the number of beats in eachpattern and label each arrow with its corresponding beat. The firstone is done as an example. • indicates a beat, u_nindicates preparation of beat. (In college, music majors usuallytake a class in conducting. They learn to move a hand or baton inpatterns like these.)5 :3\ ~22~ 31 1-a. Four Beats b.coe.f.,'-....-t-;-;l"',.\ \3:r-;-r;,. 4 1 ·2c. d..-, .....--:-'" --'....----. ('\..~..\~,,. :,"'.:--, '"\\~ .:~ .("""'\1 1.g. h.r:-.~,II. The following piece of music is in ~ time (C means ~) so each measureget~ 4 beats. You be the conductor and have someone else play themUSLC. Count silently as you direct. Does the musician start toplay at the right time? Did you both count the same? Hint: whencounting measures of rest in C time, count 1-2-3-4. 2-2-3-4, 3-2-3-4,4-2-3-4, • • • Overture"LEONORE No.3"L. VAN BEETHOVEN, Op. 72a( 1770·l827)288Tempol8IIf__-:FROM: Orchestral Excerpts from the Symphonic Repertoire for Trumpet, Volume 1, published byInternational Music Company. page 13.Permission to use granted by International Music Company(

EACH SYMBOL GIVES A HINT FOR A MISSING WORD, WRITE IN AS MANY MISSINGWORDS AS YOU CAN) THEN SHARE YOUR WORK WITH A CLASSMATE,It's the last inning in a0 sand the Lynwood~is dd The 0"---'"to the plate and gives the pitcher"(.'\---------it, ITgame between the Sixth Streets best hitter------------------says___________-makers.a@"She can't hit~" yells the first j: ---:man,The score[ll]_s~Ok.unwrapping acandyI,.---- "My dog is > she is ~ "But he suddenly falls bintoLfield for_about a.,...... ?""No, don't ¢me.on his face as the ball sailsa$the coach, and asks the pitcher, "HowI'm-_________pitcher, "I'll f{~ another _"ll _ "-------------------- --------Meanwhile in the stands a ~to it," says the-looking man with astrange /\ asks a - member, "Who is thatgirl on third '): ? She's a qwill you giveher'this J after the game?"And SAY TO HIS FRIEND WHEN HE SAW THE BANANA("YOU'D BETTER C~ OR YOU'RE------- !")289

Besides being fun to figure out, symbol stories give practice in recognizing symbols.Writing "symbol stories" can provide an opportunity for creative writing. Here is a listof symbols and possible corresponding words. Some of them are more obscure than others.Some can be interpreted in different ways. You and your students can probably add to thelist.# sharp t\ accent - equalb flat~ ledger of: unequalq natural 4 time4X multiply, timesm triplet .J.J tie or tied . divide'--''-' hold BI:> B flat (be flat) + plus, addII'J step C* C sharp (see sharp) subtract, minusdO,Ye', .. ··do scale 0 square~ bar greater than'): baseL angle, acute (~ treble (trouble) "-angle, obtusee solo (so low) .6 try••••anglet+JIT piestaff 0 circle-~ measure {}set 6- rightI rest or eighth IMCJm. metre (meter)0degreenote or eighthparallelJ-..J note or quarter A5 ray~ graphJ ..... rest or quarter % percent (present)A'D line-...- rest or whole mph,gl"",,3 rate0 note or whole n intersection .::\1 primeq~ accidentals U union( ) parenthesis (parents see)sphere (severe)=290grand staff;f@"- approximately rv similar"-~ congruent((..

BACKGROUND:Motions in geometry can be illustrated in music.REFLECTIONSAn object is reflected along an axis so that a mirror image is produced.sometimes refer to reflections as "flips.1f.:..:.• •, Students1.···1···.1The axis seems to bisect the picture created by the two objects.possesses line symmetry.The total pictureTRANSLATIONSAn object moves along an axis to a new position."slides,,"These are sometimes called..ROTATIONSAn object revolves about a point a given number of degrees.as Ilturns,,"-, -- ,\,, " ~/ ../f'-:.I\ , "..,,Students remember them,x i •291

(Reflections do occur in music, both accidentally and by the design of theIn a reflection a shape is reflected across an axis so that a mirror image isStudents sometimes refer to reflections as "flips." The axis seems to bisectture created by the two figures. The total picture possesses line symmetry.composer.produced.the pic-GV~ L17~-------------0 00 0C?01~ 011000000 0Have students reflect shapes about an axis:They could also draw in all lines ofsymmetry or reflection for shapes on ahandout page and pick out letters that areline symmetric. Words like CHOICE and MOMpossess horizontal or vertical line symmetry., ,,,d " '.,:',, ---',I,,~------__1 J ~~~--, I ,\ I /... I ,/......- --~_,i____ .., ,(Many sets of notes are not reflected within a single measure, but over many measures,,Extension:Ask students to look at or listen to music and pick out reflections. Suggest pianoor guitar sheet music of current pop tunes. (292

I. • .. .]MDVb:MbNT~IQ~T{(~((({««{Each side of this dotted line is a "mirror image" ofthe other. The last two notes in the measure arereflections of the first two notes.Finish reflectingthese notes:~...~..~..~J I~··~··§.~2. Reflect the following sets of notes in the next measure. Use the dottedline as the line of reflection.~..~.-~ll~..~..~l ~I·.-~.~3. Draw the line ofreflection foreach mirror imageseen in a measure.(Disregard thestems on notes.)4. Study each measure of this march. For each measure, draw the line ofreflection if one exists.5. Reflect these notes across the dotted line.This music can be played starting at the beginning--or starting atthe end. ~ou might like to write a melody that can be playedstarting at the beginning--or the end.293

Background for Student Page:(Students should experience translatingshapes along an axis to see that the shapevarious shapes. Have them slide geometricremains the same in the process.'---------'----.'--------'They could also performa series of translations:(Again, have students bring in music that shows repetition of various phrases. Ofall the motions, repetitions (sideways translations) are the easiest to pick out bylistening.Sometimes it is necessary or desirable to move all the musical notes up or down onthe staff so the music will fit the pitch range of a singer or instrument. To make themusic keep the same melody, flats or sharps must be added. This is called transposing.A good discussion could evolve if students share their knowledge about how music iswritten for different instruments. For example, these notes will all have the samepitch when played on the respective instruments.•1\IJ " -,/)'Use:294If r \•/\.. LIPIAlVO CLAR!tvET e.t\ICOPl-lOlVE TROMBOI.lE"(AL'O )If possible, the notes on the following student page should be played on tape orinstrument.r(

This set of notes has beentranslated or repeated inthe second measure.!;~[DNDMDV~M~NTTR~N~L~TIDN~Each space and line iscalled a "place." Thesehave been translated ortransposed up two placesin the second measure.1 I~--I----.~ .1. Finish translating or"sliding" these notesdown one place.I---- __ I~--~~2. Translate each measure into the next using the given direction andnumber of places-- I- - -- I 1- -- -1-Up 2 places Down 3 places3. Finish translating (transposing) this tune up three places.I$t~1$4 f iit$J I I I4. Notice that a b (flat) was placed on the B line. Play or have someoneplay the translation using a BU and then using a B~. Why is theflat necessary?Sometimes entire songs need to be translated because of the pitch rangeof a certain instrument or the voice range of a singer. If you play aninstrument in band or the guitar, you could explain to the class how youtranspose music for your instrument.295

(Background for Student Page:The notes on the student page have been rotated 180 0 or a half turn about a givenpoint on the staff. This may be the most difficult motion for students to "see." Itprobably occurs in written music only by coincidence.Students should experience rotating many geometric shapes in a variety of ways.The simplest for them to visualize is a rotation of a figure about a center point.While they are doing these, discuss the amount of turn taking place.Examples:leoThe center of rotation can also be a point on the figure.be produced by students.Examples:Interesting designs can(Extensior,Polar graph paper can be used to create lsome nice designs. Shade some regions in 4(or another fractional part) of the graphpaper. Repeat the design in all parts byturning the design about the center point.Ask students to find pictures of flowers or plants where an illustration of rotationis apparent. Sunflowers or other spiral designs in nature are good for this purpose.(296

TWIRD MDVbMbNT~T~Each of thesemeasures havemore examplesof half-turns.Notes could be rotated about a point. (In music thisis called "inversion. ") This note has rotated aroundthe point marked X one-half turn. The new note is thesame number of places below the X as the original notewas above the X.1. Rotate these notes one-half turn (or 180°) around the given point ineach measure:I ..... ~- .....i x 'i )(--.. -x• •I I Ix• x2. Put in the point of rotation in each measure (the turns are all 180°):3. What would happen if you rotated a note 360° about a given point?4. Put an X above eachletter to show theplace on the stafffor a note with thatletter name. Thechart below willhelp you.Gc5. In each measure above, rotate the note 180° around the point youmarked with an X.297

Background for Student Page:(Combinations of motions to compose music or designs is an interesting and challengingactivity.Here is a chance to discuss the "glide reflection" in geometry. This is a combinationof a translation and a reflection. Have students try this with a variety of shapes.First translate the figure, then reflect the image you get from the translation.Example:Experiment with the combining of other motions.it along a vertical axis. Rotate it 90 0 and reflectoriginal shape. If it is not identical, what motionSelect a geometric shape. Reflectit again. Compare the result to theor motions would make it so?Try various combinations with different shapes to find a specific set of combinationsthat will always give you your original figure back in the same spot.(Extensions:Those students interested in writing music could do so in several ways: Start with aseries of melodic notes, add measures by translating, reflecting, rotating, or combiningthe motions in a set pattern. One pattern could be established by throwing dice.The TWelve Tone Method givesan introduction to Schoenberg's method of composition that uses ideas similar to geometricmotions.Another extension of motions is tolook at the art work of M. C. Escher.If any of his books or poster printsare available, show them to students asexamples of the different motions. Seeif they can pick out the motion.Students might also be able toidentify different usage of these motionsin a marching band.(298

FINRLE:VQOIQTIDNCIt would be interesting to write music by using the motions.... - ... ...... ...... I- ... I- f ...... ....Given Reflection Translation Rotation ofof given of 2nd 3rd measuremeasure (180° about(up 2 places) the note B)Here are 3 measureswritten, each showingone of the motions .1. Identify the motion used from each previous measure:.. .G cr.·..... .... ..•........ .... ...G-........2. Finish.the tune with these motions and axes using each previousmeasure:,~... ·~... I~~~~~~~~r~t~,IGiven Reflect Translate Rotate Translate Rotate(down 2 (180° (up 5 (180 ° aboutplaces) about G) places) upper C)3. Finish this tune by using the given measure and the given motionsand axes:Given Reflect Translate(up 2 places)Rotate(180 °about C)ReflectRotate(180°about B)4. Find examples of motions in music that you or your classmates have.Look for reflections, translations and rotations.299

I. Water MusicMaterials:Activity:1. Tap one of theis half full.~IMPL~MU~I[HLINC;TQUM~NTC;8 tall identical glasses,graduated cylinder, pianoempty glasses. Now put water in the glass until itTap the half-full glass. How is the sound different?(2. Play the C scale on the piano. Fill the glasses until the 8 glassesmake the same pitch sounds as the C scale. (With some size glassesyou won't be able to get all the pitches.)3. Measure the water in each glass and record the number of millilitresof water for each pitch.cD E F GAB C_:ml t _m\ t _m\ t _m\ t__ml _._Yn\ __m\ __m\(4. Try the same experiment with another size of cylindrical glasses.Do you see any patterns in the number of millilitres of water?II.Sliding MusicMaterial:Activity:One dimestore "sliding whistle"SUDE:---1--'~-,--------

Materials: Written music with a definite melody line--chosen by studentSquared grid paperActivity:I. Pick out the melody notes of a favorite piece of music and record thenumber of times an A occurs, the number of times a B occurs, etc.Record your information in a table like this: Make a bar graph of yourresults.MOTEA B~ B C D" D E:'" E: F FI\- G A"A'" C~ D"lf G" GlIMO. 01' IIM\;'';;II.Which note occurs the greatest number of times?Which notes occur the second and third greatest times?andWhich notes do not occur at all?Try to explain whysome notes occur more often than others.III. A table is given below. See if you can guess in which key the correspondingpiece of music is written.Key oftJOTEA. B" B c D" D E" E F F'* G AI>D"IJO. OF TIMe:. 3 0 3 210 0 17 4 20 0 1 1 0IV.The table above was made from the first part of Scott Joplin's "TheEntertainer" (excluding the first 4 measures of introduction) whichis the theme song for the movie "The Sting." If you can find a copyof the sheet music for "The Entertainer" make a table for each of theparts of this piece of music. How do the tables differ? Can you tellthe part that is in a different key by looking at the tables? (Youcould try this same method on a piece of music of your choice. Be~>ure i.t hC\s several parts--at least one of which is in a different key.)301

,.(Many people recognize a composer or singer whenthey hear the music played or sung. There's somethingfamiliar about the melody or its rhythm that makes itidentifiable. Can composers be identified by somemathematical structure in their music?Some components of music that are easy to represent with numbers or number sequencesare the placement of note~in the scale and interval jumps.Most students could easily analyze a piece of music by studying the frequency ofinterval jumps.Bar graphs can be used to display the information obtained.Activity for Students:Assign the numbers I to 88 in ascending order to the 88 notes on the piano.half-tone interval is given a new number in this manner:Each(The melody line of some piece of music is then changed to a sequence of numbers.Beethoven's "Minuet in Gilis numbered here as an example:·~·F~302

The next step in the process is most quickly done by two people. The first personlooks at the numbers representing two consecutive notes, and reads the difference to asecond person who records the difference. The interval jump is considered positive whenthe second note goes up and negative when the second note goes down in pitch. The firstinterval of the "Minuet," 51 to 52, would read as a positive 1. The next interval readis 52 to 54 or positive 2. Students could record the number of each kind of intervaljump in a frequency table like this:IIllTERVI\LDII'FEREl\\CErREQUE:\\lC'I-9 -8 -1' -11. -5 -4- -3 -2 -1 0 1 2 '3 4 5 G l' 8 9The data is much more interesting when put into a bar graph.after 64 measures of tabulation:This frequency occurs'75 _50 _2.5 _NE.GATIVEPOSITIVE.DoQuestions about the bar graph could be asked:the patterns occur in another piece by Beethoven?Are there any patterns in the graph?Other graphs could show the number of occurances of each given note on the scale.Extension of Activity:More graphs could be done showing the same data about another piece by the samecomposer and compared. Other comparisons could be done on different composers, differentstyles of music, or different times in history.Question: Can we get a computer to help in the process? See the advertisement onthe next page.303

AdvertisementSolving Musical Mysteries by ComputerPbgiarism didn't concern sixteenth~century ItaJi,ll1 song \\Titers. The~' didn'tthink twice .'bout borro\dng etlchother's best tunes. And nobod\' minded.That's the way it was back then.But today, rnusico'logbts want to tnKcprecisely \dlO bOlTowed what fromwhom. And the computer is helpingthem do this, just as it is helping themin man\' other kinds of musical research.Th~ two SCOres below show how anobscure musician, :\'icola Broca, borroweda melody from a better-knowncomposer, josquin des Pres. !\icolaeven went so £

BACKGROUND:»LJJ:::J'~:.,:: ~.:.., fJ·iJ: ~.:;. § X':,·· V······ I'·:" []] 00·" ..~! {} !r;~ :.;...:; n~ :.\:'.:; i~ i~ i! :U! ({[...... ~:.... .' ...:. '::;:. :.... :~:.'. .: ....:. ' .. '.. :...Tunes can be composed using familiar number sequences for themes. Possibilitiesare Fibonacci numbers, Lucas numbers, square numbers, triangular numbers, primes, evens,multiples and factors. (The booklet Fibonacci and Lucas Numbers in the Houghton MifflinEnrichment Series gives interesting properties of Fibonacci and Lucas numbers.)The process involves writing down the sequence of numbers and converting it to noteson a scale. Let's use the Fibonacci numbers as an example:I, I, 2, 3, 5, 8, 13, 21, 34, 55, 89,Each number is found by adding the previous two numbers. (1 + 15, 3 + 5 = 8, • • .)2, 1 + 2 3, 2 + 3 =The first five numbers of the Fibonacci sequence could easily represent positionson a given scale, but what should be done about the numbers 8, 13, •••? Most notesused in songs belong to one of seven places on a major melodic scale. For example, ifwe wanted to write a song in the key of C, most of the notes would be included in theset of C D E F GAB. The symbols or letters could also be assigned numbers accordingto their position:CDEFGAB1234567Since most number patterns use more than the first seven numbers, we need to find asystem that will convert any given number to a number less than 8. Starting on C in thekey of C, the first and eighth notes are both C, the second and ninth are both D, and soOn because of the octave arrangement of scales. We can substitute 7 for any number thatis a multiple of 7. For any other number, substitute the remainder when the number isdivided by 7. The Fibonacci sequence is then converted to the following sequence ofnumbers ..1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,1, 1, 2, 3, 5, 1, 6, 7, 6, 6, 5,Notes are then written in the new sequence, with whatever rhythm seems appropriate.Here is one possible theme from Fibonacci:~At0Yo COULD BE WRITTEt-l AIJ OC,AVE00 HIGHER;:R LOWER. ./_~00I. 0I/ ~ till I-{( ""\ I._.- ~ I 0....... ./~ I'•• --I(fJ- -......,0, T\.n:''''E~NO,':':>~More measures can be written by extending the number sequence, by repeating a partof the sequence, by reversing the order of the sequence and so on. Motions and Musicgives more ideas that can be used to create additional related measures of music.Extension:Try composing music with 1the digits of the decimal representation of arepeating decimal like - or an irrational number like ~.~ ~ 3.141592653589793238462643383279502488409716939937510•••••••••305

(Material:RESTBlank music paperThree dice marked as shown below.C ° 0 ",0ho.lfA B C DCE r G RGT whole doited haJf '1ọovt",.ha.lf'C'1.uo.rt"rProcedure:--Place the treble clef sign and the ~the music paper.time signature on the top staff of--Mark off the number of measures you want to compose or spin adetermine the number of measures.spinner to--Roll one of the lettered die (your choice) and the time die to determinethe pitch and length of the first note.Record this on your staff.--Continue this process until the measures are full. (,.Special Rules:--If aC is rolled,The same is truethe note can be placed in either C position: ===: or-e>forother notes.--If a time is rolled that will not fit into a measure, roll a new time.If a quarter note or rest is necessary to fill the measure,need not be rolled.--A composer may decide to begin and end each composition with a C.the time dieIt looks like music but what does it sound like?or have a musically inclined friend play it for you."composition by chance?"Play your compositionHow do you like your (306

I.t:j: Question:RN ORDERL YRRRRNbtMtNTSuppose you want to write a melody using exactly 5 quarternotes and choosing from 7 different pitches: A, B, C, D,E, F, G. How many different melodies could you write?(Hints are given below.)Let's simplify the problem:a) What if you wanted to use 2 quarter notes and choose from 2 pitches,A and B?There are 4 possible "melodies." Can you write them below?A--', andb)JO~Jwrite the melody arrangements for 2 quarter notes and 3 pitches,A, Band C.--'--'-- , -- , -- , --, ,How many are there? or xrHow many melody arrangements for2 quarter notes and 4 pitches? or x2 quarter notes and 5 pitches? or x2 quarter notes and 6 pitches? or x2 quarter notes and 7 pitches? or x2 quarter notes and 12 pitches? or xJII.Try the same reasoning for 3 quarter notes.a) 3 quarter notes and 2 pitches. The melody arrangements are--------,------ , ------,,---- , ------There are or x x arrangements.b) How many melody arrangements for3 quarter notes and 3 pitches? or x x °03 quarter notes and 4 pitches? or x x °03 quarter notes and 5 pitches? or x x3 quarter notes and 6 pitches? or x x3 quarter notes and 7 pitches? or x x3 quarter notes and 12 pitches? or x xIII. Try to answer the question at the top of the page.here:Write your answer307

THETWtLVt- TONEM[T~OO(1.All twelve tones of the chromaticscale are used in the twelve-tonemethod of composition. Each of thetwelve tones must be used beforeany of the tones can be repeated.The sequence of twelve tones is then varied in ways somewhat likemotions in geometry. Suppose the following ordered row of twelvetones is chosen. The same ordering of tones is indicated on pianokeys and in written music.2. G 5 3 108 7 1"'" 9 "l- 12.' 11NOie:NOTEtMIDDLE:C(II.The next twelve tones can be determined by reflecting the originaltwelve as shown below. This gives the original twelve tones readbackwards. (In music this is called the retrograde series.)ORIGINAL2. G 5 3 103088 7 l s, 9 4- 12T~ 111\\0",: '-lo,.t:MIDDLECtTHE TWELFTH NOTE BECOMES THESECOND MEASUREFIRST NOTE OFTHE FIRST NOTE BECOMES THE TWELFTH NOTE OF THE SECONDMEASURE(

III.THETW~LVt- TONtMET~DO(PAC,E. 2)The original twelve tones can be transposed (translated) up or downany number of half steps or piano keys.keys between each pair of tones remains the same.The number of half steps orhas been raised a whole step (two consecutive piano keys).The example belowopo o ~e ~o I2 1 5 3 11IV. An "inversion" of the original twelve tones can also be used. In aninversion the same number of half-steps are maintained between eachpair of notes, but the direction of pitch change is different. Theinversion of the first two notes of the original sequence is givenbelow.ORIG\IIlAL!I\lVl:RS10l-lAninterval of 3 half-steps.rfirst note isthe same.The second note is 2 half-stepshigher in pitchHere is the whole inversion of the original twelve tones.(Inversionsshould not be confused with inverted intervals in which one tone changesby an octave.)v.10 4 9 '1 8T T11T12 3T 5T1 GT~1 1MIDDLE:CThere are 39,916,800 different ways of ordering the original twelvetones (12! = 12 • 11 . 10 • 9 . 8 • 7 • 6 • 5 . 4 . 3 • 2 . 1 =479,001,600, but if transpositions are disregarded at this point,there are only II! or 39,916,800 ways.)any combination of the above variations can be used.IAfter an ordering is chosen,309

THETWtL Vt- TDNtM[T/-{OO(PAGE 3)(VI.The twelve-tone technique ofcomposition was devised byArnold Schoenberg about 1910­1920. It represents a methodof musical composition whichdisregards all of the tonalprinciples of 19th centurymusic. The music written bythis method is not "in a key"with a root tone dominating theconstruction of chords andtheir succession. The methodseems to be more mathematicallybased and might appeal more to.... '.". '.'. '.' '. '. ' .. ..'..11;>. a a• - l?a ~•(c)(d)_P.P. 1>. •-• P. p1>'._ ". 1;1_1;1 a -- v.• -• a l •( b)(310

Musical tones sound different whenthere is a change in pitch or loudness.Graphs can be made to representthe pitch, loudness and qualityof a musical tone.I. The pitch is how high or low the sound is.number of vibrations a string, vocal chord,a unit of time. The graphs below representIt is determined by thedrum head or reed makes intones with different pitches.one vibration orcycle per second2 cycles per second cycles per secondII.1/'" '"/oThe more1/\ 1/\\/ 1\/1. SEC-OlUO 0 1. 0cycles per second (cis) the higher the pitch.'\\. \ .1.Circle one:a) 5 cis has a (higher, lower) pitch than 2 cis.b) 10 cis has a (higher, lower) pitch than 100 cis.III.The loudness depends on the distance that the string, vocal chord orreed is vibrating. A gentle pluck on a guitar string moves the stringa short distance and produces a soft sound.The louder the souna,the morethe graph of the sound is"stretched" up and down./'" '".// \/1\\ /o 1.':;OFT ,:;oUNDo 1-LOUDER SOUNDIV. Compare graphs b-e to graph a. Is each pitch higher, the same orlower? Is each sound softer, the same loudness or louder?eJla.bc I \ dI I/" "- /\\1 \/I0 \ /1.o 1.o 1.o 1.o 1.pitch compared to a: b c d _ e _loudness compared to a: b _ c _ d _ e _311

( .V. Most people can tell a difference in the sound of a piano and thesound of a trombone. A note played on a piano at the same pitch andloudness as a note played on the trombone sounds different. Thesounds produced by the two instruments are said to have differentquality or timbre.VI. The shape and design of a musical instrument gives it a characteristicquality. Graphs of different notes on the same instrument always havethe same general shape. The effect of different instruments can beseen in the shape of the corresponding graph.l-Voal~NIIlGr~ORKV \\'"1 2/ '\ 1/'" I" / " /o1v- ,/ V- VVIII.IIob)1\\ l/ \ /2Sa:OIJDS~T~/'"/ rv..\1"'\ / M /Match these graphs with one of thegraphs above. Name the instrument,then give the pitch as higher, lOW-+tm#tter or same and the loudness assofter, louder or the same.o 12.illI'\ '\ III1 2.1\ 1\vlv1 IVIII \oc)v1d)(VIII. Joseph Fourier, a Frenchman born in 1768, showedthat all sounds can be described in mathematicalterms. His formulas and graphs have been veryimportant in the design of the telephone, radioand stereo equipment.(312

The chart below gives the range in pitch for voices and instruments.The frequency column tells how many vibrations per second are necessaryto create a tone with the given pitch. Look at the chart and answer theI1::-1IIe.I,GGVioli",CViolaCello A ,8011Dn"00010!C,.Oboe, , li,h Hom AD Clarinet I ,.80ss Clorinet0' Alto S(1l

-~r:Jb Wb ON TWb, !:~Mb ~Ql:OUbN[l,J ?Cyclesper secanThe chart below gives the frequencies of the tones of a piano. The frequency ofthe A above middle C is 440 cycles per second. This means any string, reed or vocalchord would have to vibrate 440 times in one second to produce a tone which wouldmatch the pitch of this A on a (tuned) piano. Use the chart to answer the questionsbelow.cOr-! 0'\~6 ",c!, '" M .,. 0'" co co M "'0'" r--", "'.-1M r!..", o~'" "'0 0.,. M'" M .,. '".,.r-!r--'" co 0'"r-!'" co '"'" co "1"'''1 "I r-- .,. . . '". '". . . ° . . . . . . . . . . co .,. Or-!'" r-- '"ON. . . . . . . . • M '" M '" '" .... r-- r-! .,. "I "'0N 0.,. co"'", r-! co'" "I'" '" '"'".,.CO "'r-!co'"'" .... NOr-! M'" COO M r-- r-!MMM r-! "I "I'" r-! '" "I.,."'CO "I .,."I MM .,. '" '" '" M r--'" r-- '" r-! r-! r-! r-! r-! "I "I "I M M.,..,.r-- co '" r-! r-! r-!r-!r-! "IN "1M M'" '"oco M COM .,. '" 0'"'" '" r-- .... '" 0 .... r-! M r-! r-!O O.,.M M MO OCO '" M'" 0 .... '" .... '"0'"° "'0 0'" ° 00 '" '" OM Or-! 00'" 0.,. co coco '" 0 0"'''' '" '" NO oco "I .,.'" ."IN or--r-! r-! '" M r-! MN '" '" M. . . . . . . . . .MO COO Or-!O"'co .... .... "1 .,..,.'" '" or-- ...::t M 0'\ . . . . . . . . . . .'" M '" r-- M'" Or-!'". . . . . . .OM° "'.,. .,.'" O"'r-! OM M M '" '" "I r-- '"co M Or-- .,. r-- r-!'" '" '" .... '" .,. M '" M "I '" co'" .,.M '" co or-! r-!r-! r-!r-! r-!r-! "I "IN "I M M M.,..... 0 "I "'r-! M co "'r-! MN .... r-- r-!N M .,. '" r-- '" "I .,. '" '" "I .,. '" '" "I co '" co co '" 0 r-!M M '" r-- '" 0 M'" .... r-! "''''r-!"1M M M .,. .,. .,. '" '" '" .....,. .,.co co '" r-!r-! r-! r-!r-! r-!r-! "INN "1M MM '" '" '"'" r-- co '"ccc~ ~ ~ ~ ~-+CMIDDLECcCC-1. List the frequencies from smallest to largest of all the C's on the piano. Whatpattern do you find in these numbers?2.3.List theChoose apattern?frequencies ofI:>black key (B ,the G's. Does the same pattern occur?b bA , E , etc.). Do the frequency numbers make the same4. A C major triad consists of a C, an E (4 half-steps above C) and a G (7 half-stepsabove C). An F major triad consists of an F, an A (4 half-steps above F) and a C(7 half-steps above F). Fill out the charts below for the C and F triads. Do yousee a pattern in the charts?I I tC E I G E+-C G+-CI'" I A c A';'!'" C';'I'"yoU MIGHT USEA CALCULAiOR TOl-lELP FII.lD PAITERNSAND DOTI-IECOMPUTATIO!..l..~G5.40G I 82.A0'7 I 97. 9991'30.812(;1.G3523.25,104G . 5 I I I k~----.J43.GS41 55.0008'7.30'71'74.(;1349.2.'31>5.400IG98.~ I I I I~,

Sound is caused by a vibrating object.The vibrating object starts nearby air moleculesmoving back and forth. These vibratingair molecules cause other air molecules tovibrate until the molecules near an eardrumare moving back and forth. The moleculesnear the eardrum cause the eardrum to vibrateat the same rate as the original object.Nerves transmit a message to the brain andthe brain interprets the message as sound.Air or some other substance is necessary tothe hearing of sound. No sound will be heardif a tuning fork is vibrating in a vacuum.The faster an object vibrates, the higher thepitch of the sound becomes. The greater thevibration, the louder the sound. The closerthe vibrating object is to the ear, the louderthe sound.1. List three of the loudest sounds you can think of.2. List three of the softest sounds you can think of.3. Compare your list with those of your classmates'. Of all the soundslisted, which do you think is the loudest? the softest?4. There are quite a few sounds that would fit between the softest andloudest sounds you just named. Some examples of sounds that peoplehear are given below. Arrange the 13 sounds in order from softest toloudest. Does your ordering agree with others in the class?food blender ~::~::;sation (close) \Al~whisper ; ~alarm clock "'-.,{-l ;. ~ -~rock band~ ~fi!r:-"'-kZ"-­q~iet ~oun~ryside ~~1'--,~\\1A"'~a~r ra~d s~ren ~. f I )t!l(\ ~heavy traffic ;;:quiet streetleaves rustling ~I I':::::: )boiler factory eU L:/Jquiet office315

~U&1nI[I§F~15g~~g5(PAGE 2)The human ear is sensitive enough to hear very soft sounds and yetstrong enough to stand.pain from very loud sounds.No one mechanical instrument canmeasure that range. It is like abalance scale sensitive enough toweigh a single human hair with reasonableprecision and strong enoughto weigh the battleship Missouri:(The decibel scale was designed to handle this wide range. It is usedon decibel readers to measure the intensity of a sound. If one sound is10 times as intense as another, its decibel reading would be 10 moredecibels.5. Compare your order of sounds to this chart. Then finish the 2 columnsusing the patterns started.SOUNDRELATIVEINTENSITYDECIBELSthreshhold of hearingleaves rustlingquiet countrysidewhisperquiet office110100100010000o10203040(quiet streetconversationheavy trafficalarm clock (danger level)subwayfood blenderrock groupboiler factory (threshhold of pain)air raid sirensound that burns skinlethal levelnoise weapon316

6. The relative intensity column is given in powers of ten. Each soundis 10 times as intense as the sound before it.Write some of the relative intensity numbers using exponents.For example: 10 = 10 1100 = 10 x 10 or 10 21000 = 10 x 10 x 10 orthreshhold of pain =lethal level =threshhold of hearing =Make a new column inthe sound chartwriting the relativeintensities aspowers of 10.7. What sound is onehearing?one billion timesthousand times as intense as the threshhold ofone million times as intense?as intense?8. Look at each sound and compare its relative intensity to its decibelrating. Explain how the relative intensity of a sound can help youfigure out the decibel rating.Dearie, you're something, that loud mouth and all!- 9.LONDON, England (AP) ­Michael Featherstone and hiswife, who run a pub on theYorkshire coast, have beenjudged the world's loudestmouths.Competing Wednesday in the"Second World Shouting Championships,'"Mrs. Featherstonelet loose a 109.7-declbel yell toretain the woman's title she wonlast year.Her husband was the loudestof the male competitors with 110decibels. But he missed the 111­decibel mark set-by the winnerlast year, a fisherman.Compare the Featherstone's records to the decibel reading on the soundchart. They each yell about as loud as a10. Read the articles on the next page.a. In the article "Loud Music May Cause Migraines,"how many discotheques would each inspector have toinvestigate if they were all going to be checked inone night?how many young people out of the 7,000 would you guesshave measurable hearing loss?b. Do you think the "caution" in the stereo article is a warning?Why?c. Do you like to listen to really loud music? If you do, youmight want your hearing checked frequently.Permission to use granted by The Associated Press317

318Loud musicmay cause• •migraInesCALIFORNIA. P". (AP) - Superloudrock music is

1. Here are levels of loudness that musicians use:symbolpppmpmffffwordpianissimopianomezzo pianomezzo fortefortefortissimomeaningvery softsoftmoderately softmoderately loudloudvery loudWhat would fff mean? ---------- ppp?2. Suppose there is an increase of 6 decibels from one level to the nextso that music played at the piano level of sound would have a readingof 6 decibels more than the music played at the pianissimo level andso on. How many more decibels would the ff level be than the pp level?_____ How 'many mOre decibels would the fff level be than the ppplevel? Can you tell how many times louder the p level isthan the ppp level?3. Guess the decibel readings for these sounds:soft recorded musicloud recorded musiclive symphony (very loud)decibelsCompare your numbers to those of a classmate, and check with yourteacher. Add this information to the chart on Numbers for Sounds.4. Two other musical symbols that control sound are-===--=crescendo(gradually increasingin loudness)anddecrescendo or diminuendo(gradually decreasing ordiminishing in loudness)What two symbols do they remind you of in mathematics?319

(Select some of the following projects and use adecibel reader:I. How loud is the music you listen to?Record the decibel reading of music as it's played on aplayer:~" .. ,glllllill::',.Qsoft (volume at background level):::..~_ medium (turned up for listening)loud (turned way up for dancing)radio or recordII. How loud do you sing or play your instrument?Record the decibel readings of individual instruments or someone singingat different dynamic levels.r------------------ ... --------------- - - ... - - - --..,: instrument dynamic level decibels:: I: soft :II: medium :IIL:...loud-':III.Can you stay at one intensity leVel?Which instrument playedthe loudest?the softest?Have a performer play at one dynamic level and watch the decibel reader.Try several types, including a brass, woodwind and string instrument.Record the name of the instrument and its decibel range as it plays up,then down the SCale. Is there a pattern? Does a tone need to belouder when it is first played then when it is held?IV. HoW does a tape recording affect sound?While recording the decibel readings, also tape record someone who playsor sings a melody at various given dynamic levels. Then play the taperecording and read the decibel levels of corresponding sections of themelody. Are the levels the same? (Leave the volume knob at a constandreading.)V. Is ·there a difference between records, TV, radio and II Iive "?(Record the decibel levels during equal amounts of time of music played ontelevision, on the radio, on a record player, and live (a group of musicians).Find the decibel range of each (the difference of its loudest andsoftest). Which one of the four ways of listening to music gives you thewidest decibel range?VI. What are the decibel rat.Lngs of sounds around your school?Guess where certain sounds would fit on the chart,reader to record the sound levels at these areas:cafeteria, bandroom, computer terminal.then use a decibelclassrooms, hallways,(320

Decibel readers are small electronic instrumentswith meters that record sound levels.Most are batteryoperated for ease in handling.There is usuallya microphone attached to pick up the sound and a dialor meter that registers the readings in decibels.To find one, check the science department or audio-visual center of a junior highor high school. Your district instructional materials center might also have one.Another source is the Noise Pollution Authority, who may have both the instrument anda guest speaker for a class. Decibel readers are also available at retail stores.Students enjoy using the decibel readers.Use of the instrument attaches some meaning to the numbers. The following studentpage provides activities in which a decibel reader can be used.Extensions of Student Pages:Have students watch newspapers and magazines for articles related to noise.TheFederal Occupation Safety and Health Standards Act and the recent Noise Control Act areresults of data that show the hazards correlated with excessive noise.Recent evidenceshows not only damaged hearing, but migraine headaches, high blood pressure, stomachproblems and other emotional disorders.321

(1) How does sound travel? Howfast does it go? How isloudness related to distance?2) What is an echo? How doesa sonic boom occur?/i3)What sounds do you like tohear? don't like? What isthe difference between nOis~ _. ~and ~us:~G~~3'I. 'fJ1~ ~ r

C£OJSA2Yaceidentals. Musical symbols likesharps, flats, naturals, double sharpsor double flats (,jj:, l? , ~ ,#'4t ,bb )which raise, lower or return to normalthe pitch of a note.allegretto. A tempo; light, cheerful;like allegro, but a little less fast.allegro. A tempo; quick, lively,rapid and cheerful.barlines. Verticle lines drawn onmuscial staves to separate measures.~~ ~~ /Barlinesbass. 1. The stringed instrument inan orchestra with the lowest pitch-­the double bass. 2. A low-pitchedsound. 3. The lowest part in 4-partharmony.bass clef. The bass clef is oftencalled the F clef. The symbol startson the F line and the : is centeredon the F line. The F below middle Cis written as follows.t=7-=--:_0__brass instruments. Instruments liketrumpets, cornets, tubas, trombonesand French horns which are made frommetal flaring into a bell. The vibrationsof the player's mouth areamplified by the instrument. Valvesor a slide on the instrument plusthe tension in the player's mouthdetermine the pitch of the tone.chord. Simultaneous sounding of th~eeor more notes.chromatic scale. A scale consisting oftwelve half tones to the octave (or 13if the first note is repeated anoctave higher).conductor. A person who directs a bandor orchestra. The conductor determinesthe tempo, the loudness and ingeneral how the piece is to be played.crescendo. A gradual-=======increase in loudness.Symbol:decibel. A unit for measuring loudnessor intensity of sound. A decibelscale runs from about 0 decibels forthe average.least perceptible sound toabout 130 decibels for the averagepain level.decrescendo.loudness.A gradual decrease inSymbol:dotted notes and rests. A dotted noteor rest has one and one-half the timevalue of the note or rest. Examples:d.J.has the time value of a halfnote and a quarter note.has the time value of a quarternote and an eighth note.-...-. has the time value of a wholerest and a half rest.duet. Two performers. Music for twoperformers.eighth note. A note which receives onebeat in ~ time. It has one-half thetime value of a quarter note.Symbol: Jeighth rest. A rest withvalue as an eighth note.the same timeSymbol: '1electronic instruments. Instrumentswhich produce sounds which are electronicallygenerated. A Moog synthesizeris one example.acc-ele 323

MUSIC GLOSSARYfif-notfifth.firstscaleflat.beforeshouldforte.fortissimo.A perfect fifth consists of theand fifth note in a major(an interval of 7 half tones).The symbol (!? ) which is placeda note to show the pitchbe one-half step lower.Loud (f).Very loud (ff).gold record. A record given as areward to a performer or group of performersfor selling one millionrecords or a million dollars worth ofrecords.half note. A note which receives onebeat in ~ time. A note with twicethe time-value of a quarter note.Symbol: dhalf rest. A rest with the same timevalueas a half note. Symbol: ~half step. Located on a piano by goingup (or down) from one key to the nearestblack or white key.half tone. A half step. The intervalbetween two adjoining keys on a pianoor two adjacent tones on a chromaticscale. Synonym: semitone.hexad. A six-note chord--usuallycalled a hexachord.intensi1;y • 1. Informal--relativeloudness of a sound. Feeling ofsoft to loud. 2. The amount ofpow~r per unit area.key signature.\ JI"'DSharps or flats( :It, t1) placed after the clefsign to give the key of thepiece of music.B P key signaturemajor scale. A series of 8 notes withwhole steps between consecutive notesexcept between the third and fourthand the seventh and eighth whichdiffer by half steps1/2 1/2C major scale: C,D,~,G,A,~F major scale:measure.lines.1/2 1/2F,G,A:B ,C,D,E:FThe space between two barone measuremelodY. An organized succession ofmusical tones.metronome.exact timetick and aAn instrument which marksby a regularly repeatedregularly swinging arm.mezzo piano. A level of loudness.Moderately soft (mp).mezzo forte. A level of loudness.Moderately loud (mf).monotone. A single unvaried tone.ing on one tone.natural. Indicates that a basic toneis no longer sharped or flatted.Symbol: qnon-pitched.Sing-Without a definite pitch.note. A symbol used to give the pitchand duration of a musical tone.0 whole noted half noteJ quarter noteJ eighth note I((324

MUSIC GLOSSARYoct-stroctave. An interval consisting of 8tones from a major scale (12 semitones).An interval from one C tothe next higher C, from one G to thenext higher G, etc.quintet. A group of 5 performers.Music for 5 performers.rest. A symbol used to indicateperiods of silence.octet. Music for 8 performers. ..-r whole restopera. The musical form of drama.The drama is set to music for voicesand instruments.overture. 1. An introductory compositionfor instruments. 2. A prelude to anopera or oratorio.percussion instruments. An instrumentlike a drum, xylophone or trianglewhich is s truck, shaken or scraped.pianissimo. A level of loudness.Very soft (pp).piano. A level of loudness.Soft (p).pitch. The relative highness or lownessof a tone. Pitch is determinedby the number of vibrations per second(frequency).prestissimo. A tempo.as fast as possible.tempo in music.Extremely fast;The fastestpresto. A tempo. Very rapid; fasterthan any tempo except prestissimo.promoter. A person or group who hasthe financial responsibility of amusical or sports event, includinghiring the musician, renting a placeand advertising the event.quarter note.one beat inis .Iquarter rest.A note which receives~ or it time. Its symbolA rest with the sametime value as a quarter note.Symbol: ~quartet. A group of 4 performers.Music for 4 performers.____half rest~ quarter rest1 eighth restseaZe. A series ofwithin an octave.and major scales.rising pitchesSee chromaticsextet. A group of 6 performers.Music for 6 performers.sharp. The symbol (~) which isplaced before a note to show thepitch should be raised one-half step.sixteenth note. A note with onefourththe time-value of a quarternote. Symbol: ~sixteenth rest. A rest with the sametime-value as a sixteenth note.Symbol: ::tsoZo. One performer. Music for oneperformer.staff. Five parallel lines on whichmusical notes are written.standard pitch.per second isa' = 440 vibrationsthe standard pitch.string instrument. A musical instrumentlike a violin, harp or pianowhich produces a sound when a stringis plucked or struck or when a bow ismoved over the string.325

MUSIC GLOSSARYtern-wootempo. The rate at which a musicalpassage or piece is played. Thetempo may be given by a word such asallegro or presto, or an exact metronomemarking.third.firstscaleAn interval consisting of theand third tones of a major(an interval of 4 half-steps).timbre. The quality of a tone. Timbreis determined by the prominenceof overtones--other softer pitcheswhich combine with the main tone.time signature. A symbol indicatingthe number of beats per measure andthe type of note which gets one beat.The symbol 4 tells there are 3 beatsper measure and a quarter note getsone beat.tone. A sound of definite pitch,loudness, timbre (quality) and duration.It is made by a regularvibration of an elastic body.transpose. To write or play a piece ofmusic in a different keY.treble. 1. The highestmonic music: soprano.pitched.part in har­2. Hightrio. A group of 3 performers. Musicfor 3 performers.triplet. A group of 3 notes of equalvalue to be performed in the place oftwo notes mof the same value.The triplet would beplayed in the time of~ 2 eighth notes or 1quarter note.l;~~whole note. A note which has fourtimes the time value of a quarternote. Symbol: C)whole rest. A rest with the same timevalue as a whole note. Symbol: ~woodWind instrument. An instrumentlike a flute, clarinet, saxophone,oboe, bassoon or English horn whichconsists of a wooden or metal tubewith finger holes or keys to vary thepitch. The sound is created by one ortwo reeds vibrating in the mouthpieceor by air passing over an open hole asin a flute.treble olef. The treble clef is oftencalled the G clef. The symbolcrosses the G line 4 times. The Gabove middle C on the piano iswritten on the second line of thestaff as shown:/.rr'- LJ """Gtriad. A chord consisting of 3 notes:the root note, a third and a fifth.326

to MATHEMATICS ANDPHYSICSNatural interests for students' investigation is theirimmediate environment and the physical forces that affecttheir lives. Gravity on earth is a force that determinesAIRtheir weights and restricts their actions; air exerts pressureon them, and they must use various forces just to move, playand work. Light and optical phenomena directly influence theway they perceive their environment. These are some of thetopics that are studied in physics, and many of these ideascan be introduced to students at the middle school level.The study of physics utilizes mathematics from simple measuring and graphingtechniques to abstract algebra and differential equations. In this section you willfind several data collecting and graphing activities that provide information aboutthe strength of magnets, the action of a pendulum, and the forces needed to moveobjects on an inclined plane. Computational skills are needed in studying the speedof light and the half-life of radioactive substances.Ideas about area and volume are used to calculate air E.o....thpressure and specific gravity. Proportions are usedin studying bicycle gears, levers and inclined planes.Symmetry and other elementary concepts from geometryenter the study of optics, in particular, light reflectionand the way mirrors work.The lessons included here provide many opportunities for student involvement.A singie scientific phenomenon is explored with students using mathematics duringthe investigations. Students will be active participants during the lessons ratherthan simply doing a worksheet about the activity. For those lessons involving unusualor more extensive scientific study, teacher pages are provided to supplybackground information. An effort has been made to be scientifically correct inthe use of all terms having specific scientific meaning, such as mass, weight, pressureand force. Only when it would be confusing to the students is a change made.For example, grams per square centimetre are used for pressure instead of newtonsper square centimetre. All of the material needed to do an activity are easy toobtain or make, e.g., rubber bands, paper clips, graph paper and weights.The pages in this unit can be used in a variety of ways: to review particularmathematical skills, to connect mathematics with a physical phenomenon, to involvestudents in actual experiments, and to encourage individual research and discovery.327

INTRODUCTIONMATHEMATICS AND PHYSICSYou might prefer to question the students about their interests related to physical (phenomena and then design or have them design an investigation like those suggestedhere. Students could work individually or in small groups on differentactivities and then put on demonstrations for the class.It was not feasible to cover all the topics of physics or even touch on eachof the areas. Only samples are provided here. You will find activities related toForce that measure the strength of a magnet, calculate air pressure, use propertiesof levers, and determine the strength of a rubber band. Included are activitiesrelated to Work that deal with inclined planes and the effect of gravity. Notionsabout Heat are involved in two activities using a thermometer. There are graphingand computational activities dealing with the half-life of radioactive particles.The topic of Optics is touched upon in activities that deal with the speed of light,symmetr~ and reflections related to mirrors. Activities on Sound can be found inthe MATHEMATICS AND MUSIC section of this resource, but they could also be used ina unit on Mathematics and Physics.We have a wonderful opportunity to involve students in activity-oriented lessonsin physics that help them appreciate the physical forces that affect theirlives and, at the same time, use the mathematics they have learned in meaningful (applications.(328

MATHEMATICS AND PHYSICSTITLEPAGETOPICMATHTYPEThe Meeting of theNorth and South331 Measuring strength ofmagnetsCollecting data andaveragingActivity cardManipulativeThat Heavy Air332 Calculating air pressureFinding areaActivity cardDemonstrationTeacher pageA Swinging Time334 Working with apendulumCollecting data andgraphingActivity cardManipulativeRolling Bodies336 Timing balls rollingdown an inclined planeCollecting data andgraphingActivity cardManipul,itiveHang Ten337 Measuring strength ofrubber bandsCollecting data andgraphingActivity cardManipUlativeProportions With aPlank339 Working with a leverWorking with proportionsTeacher-directedactivityGet in Gear 340 Working with bicyclegearsInclined Planesin Use 341 Examples of inclinedplanesUp the Ramp 342 Working with inclinedplanesBeing in the Middleof Things 344 Find center of gravityWorking with proportionsAwareness of how inclinedplanes are usedCollecting data andproportionsMeasuring and definingtermsTeacher-directedactivi·tyBulletin boardTransparencyActivity cardManipulativeActivity cardManipulativeIs 32 Degrees Hotor Cold? 345Working with celsiusthermometerCollecting data andgraphingActivity cardManipulativeA ChillingExperience 346Working with celsiusthermometerCOllecting data andgraphingActivity cardManipulativePossibly So Columbus 347Calculating the distanceseen on a curved surfaceUsing a formulaWorksheetVolume byDisplacement 348Showing equal volumesdisplace the same amountof waterVolumeTeacher idea329

TITLEPAGE'fOPICMATHTYPEArchimedes Knew349Calculating specificgravityProportionActivity cardManipulativeDecay and Half-Life350Working with radioactivedecay and half-lifeCollecting data andgraphingActivity cardManipulativeTeacher pageHalf-Life353Working with half-lifeof radioactive substancesPracticing compu·tationalskillsWorksheetFaster Than aSpeeding Bullet354Working with the speedof lightPracticing computationalskillsWorksheetReflections356Using mirrors to findsymmetry linesSymmetryWorksheetMirror, Mirror onthe Wall358Working with the law ofreflec·tion of lightMeasuring and collectingdataActivity cardManipulativeImages 359Mul·tiple Reflections 361Working with imagesusing a plane mirrorUsing two mirrors toproduce multiple imagesMeasuring and collectingdataMeasuring and collectingdataActivi·ty cardManipulativeActivity cardManipulative(Reflect on ThatImage362Using a concave mirror toproduce imagesMeasuring and collectingdataActivity cardManipulativeTeacher pageBending of Light:Refraction365Working with the refractionof lightMeasuring and collectingdataActivity cardManipulativeDemonstrationTeacher page330

SUPPORTMaterials:Metre stick,2 magnets, rubber bandT~l; Ml;l;TING 01-=./VORT~A"lDSGUYWStudents will measure the strength ofmagnetic attraction using a nonstandardscale. They will alsoinvestigate how the force ofattraction is changed whenpaper is placed between themagnets.PULL UNTIL Tl-1E.TWO MAG"lE.T5 COMEAPART.ITape a rubber band to one end of a magnet and arrange in the starting position asshown.Lll\IE UP THE EIJD OF~'s c TI-iE FIR'5T MAGl-.lET WITHIV ., THE 0.0em MARK 010THE METER STICK.Cutting the rubber band and using it as asingle strand allows for a longer stretch.STARTI"lG POSITIOJ-JIIPlace the second magnet in contact with the first and slowly slide the combinationbeside the metre stick as shown in the top picture.Make ten trials.magnets separate.Each time record how far the rubber band stretches before theFind the average of the ten trials.Length ofStretches1) 3) 5) 7) 9)cm cm cm cm cm2) 4) 6) 8) 10)cm cm cm cm cmAverage stretch.___ cmIII Place a piece of paper between the two magnets.with the paper separating them.sk{APERC\==Jlj/L ~ M¥ !(IILength ofStretchesAverage stretch.Make another set of ten trials1) 3) 5) 7) 9)cm cm cm cm cm2) 4) 6) 8) 10)cm cm cm cm cm___ cmIV Instead of paper, try a small piece of metal. Again take ten trials and computethe average.Length ofStretches1) 3) 5) 7) 9)cm cm cm cm em2) 4) 6) 8) 10)cm cm cm cm cmAverage stretch.____ cmWhat effect did these materials have on the magnetic attraction of the two magnets?331

Above every person is a tall column of air pushing downwith an amazing force. This causes air pressure.A column of air extending through the atmosphere havinga cross sectional area of 1 cm Z has a mass of about 1,034grams. So the pressure caused by air is 1,034 g/cm Z atsea level. (14.7 pounds/in Z ),,I, , , ,, ,l"isQ-"\''''''' '("IIISeveral simple classroom demonstrations to show this pressure are outlined below.TII\lCAt\)HOT. ..:\;f;:./:: STEAMPLATE.-.AIR PRESSUREBE. SURE. II-\E P"'I"ER ISSMOOTHED COWl.] WITH NOAIR pOCKETS UNOE.RNE.A'H.Put about 2 em of water in a rather large metalcan that can be sealed with a lid or rubber stopper.Place the open can with water on a hot plate untilthe water boils for several minutes.This willdrive out most of the air leaving only steam in thecan.Remove the can from the heat and quickly sealit. As the can cools the steam condenses, creatinga partial vacuum inside the can.With little airpressure left inside the can will be collapsed from (the outside pressure.Cover a wooden slat (about the size of a ruler)with a large sheet of paper.newspaper is sufficient.A layer or two ofLet about 5 em of thisslat stick out from under the paper and over theedge of the table. Smooth down the paper. With ahard sharp blow strike the exposed end of the woodenslat.Due to the air pressure above the paper thewooden slat will break without tearing the paper.IIIL-/_~/" :".... :.:" .Completely fill a glass with water.Carefully place a heavy piece of smoothcardboard on top the glass of water.Hold the cardboard and invert the glassof water.Keeping a hold on the glassstop supporting the cardboard.332 By calculating the surface area involved students cancalculate the air pressure in each of these cases.It willmagically seem to hold the water in theinverted glass.GLASS H[LOIt\) AIR WA'ER It-lSID£GLA~"'> E'i.E.RTS. ..... '. . .. A. DOWI\l\.JARDFORCE.~ AIR PRESSORE ONT THE. OUTSIDEEY.ERTS A GREA,ERUP\.JARD FORCE.(

AIRThe vast ocean of air above us exerts pressure similar tothe pressure you feel from water when you are swimming.Scientists have shown that at sea level this air pressure is1,034 grams per square centimetre.1)What is the total force (in grams) due to airexerted or the top of the 2 cm x 2 cm squareto the right?grams, , , ,-----t------, , ,T\.1ERE. \.0\LL BE1,0349 DtERT£DON EACH SQUARECEt-lTIMETRE.2)When a can is open there is as much air pressure on eachsquare cm of surface inside the can as outside. If the canis sealed and the air pumped out, the air pressure inside isgreatly reduced. Calculate how much force due to air pressurewould be on each of these sides.LENGTH WIDTH AREA FORCE DUE TO AIR PRESSUREFRONTSIDETOP3)Your lungs make up a large air cavity in your body.Think up a method to measure the area of your backin square centimetres. Calculate the total force onyour back due to air pressure.____________ area of your back_____________ force due to air pressureHow come air pressure does not- crush you?333

Materials Needed:ISet up a pendulum as shown.string length of 50 cm is a goodstarting length.RSWINGINGTIMEMetre stick,about a metre of string, paperclip, weights (fishing sinkersor metal washers), watch with asecond hand.AA simple pendulum can be made byhanging weights on a string.9~~"~~~~BBJT 00,i WEIGHTS HUNGPAPER o,. ON PAPER CLIPCLIP o o ~ 0°00(O~E BEAT ,~ Pull the weight back about 10 cm and release it.A COMPLETE SWIN

-=-RSWINGINGTIME(COtJTII\lUED)III Use the same number of weights as in trial #1, butshorten the pendulum 10 cm.What effect do youthink this will have on the time for 10 beats?Try it.Record your results in the table below.StringAmount Time for Time forChanged Length 10 Beats 1 BeatTrial 1/5Trial 1/6Trial 1/7Shorten by10 cmShorten by10 cm moreShorten by10 cm more40 cm30 cm20 cm10 cmTrial 1/8 longer than 60 cmtrial 1/1Trial 1/9Increase10 cm more70 cmC~AtJGE.THE:U:k\GTHOF T\iCPE~DULUMA. LITTLEYour ownTrial 1/10choice---IV Make a graph of trials #4 through #10.A seconds pendulum has one beat inexactly one second. Using yourgraph what would be the length ofa seconds pendulum? ___ cmHow can you be Sure this answer iscorrect?\0 20 30 '10 50 C?J:) '10 80LE.fJGTI-\ Of 5TRIt-lG It\.! om.335

(Materials Needed:2 metre sticks, smooth rubber ball, watch with second handIAssemblyTape two metre sticks togetherto form a V-shaped track forthe ball to roll down.eMA smallpiece ofcardboardtaped tothe top endgives someadded support.II Set upBraces onthe side ofthe trackkeep itfrom turning.THR££ SECOI0DS,STOP!Raise the top end of the V-shaped track 1 em.III Working with a partner, find how far down the V-shaped track the ball will rollin the specified times.Time the balla s 1 s 2 s 3 s 4 shas rolled(Distance theball traveleda cmIV&;e 2.00"""0~@.J~W

Materials Needed:paperHang a rubber band with a bent paper clipfrom a support to act as a spring balance.3-••,•••~After the setup is complete add the identical weights ;one at a time and record the total stretch of therubber bands each time.Two identical rubberbands, about 20 identical weights (suchas fishing sinkers, machine washers ornuts), metre stick, paper clip and graphISUPPOR.T A Iv'\ETR.£.STICK- N£.'n TOTHE SET-UP TOUSE AS ASCALE.o.:. ... : 51.! PPQRT::·:...BEI,n PAPER eLlPACTS BOTf.! AS APOII\lTE:R AtvD AWEIGHT HAtvGER.A G.R.APH OF TI-11SIl\lrDRMAilOt-lWOULO LOOK.~IIv'\IL"'R. TOTHIS.Number of weights hung°1 2 3 4 5 6 7~Total stretch (cm) 0 (IIHang two identical rubber bands over the support with the paper clip suspended.. : : .:..' ',:. from both of them as shown. As in part I add theweights one by one and record the total stretch of therubber bands.Number of weights°1Total stretch (cm) 02 3 4 5 6 7 8 9III Change the two rubber bands so they aretied together, hanging in tandem.Will this produce a different stretchthan both parts I & II?Add weights one by one to find out.Number of weights°1 2 3 4 5 6 7 8 9 10L...T_o_t_a_l_s_tr_e_t_c_h_(_c_m_)-L_0-L_-L_-'--_-'-_l---''---'_-L_-'-_-'--_\337

IVGraphing the results.(COl\llIlJUEO)(98J: .; ",,0°0°Ust-'"4O!STRETCH OF TH£RUBBER BAND 11\\CENTIMEl"R£5o'"I-JUM8ER O.WEIGHTS I-\UIJGoO'--:-i;,-;;,,-:;~:-i5'---;G,--J7'-:8'-:­WEIGHTSOn a piece of graph paper mark the horizontaland vertical axis as shown.Separately enter the results from parts I, IIand III. A different color for each one makesthem easy to tell apart.Do you see any similarities between the 3parts?tn' 2:':n.t ';',"c1:',

PLANKMaterials Needed:Long plank about 3 metres long, concrete building block, bathroomscale, measuring tape, metre stickI. Balance the plank by placing the block in the middle. Ask for a volunteer (orthe teacher) to stand on one end of the plank. Have different members of theclass try to balance the plank by standing on the opposite end. For the plankto balance students should realize the weights of the volunteers should be aboutequal. Weigh the volunteers.II. Pick two members of the class having different weights. Weigh them and recordthe weights. Keep the block in the middle and ask them to stand on oppositehalves of the plank and balance each other. Students will probably use theirprevious experience with teeter-totters to accomplish the task.III. Again pick two members of the class having different weights. This time theirtask is to move the block so the plank will balance with one of them on eitherend.IV.Have the students use the three activities above to formulate a conjecture abouthow to make unequal weights balance. Students will probably say that the heavierweight is closer to the block, and the lighter weight is farther away from theblock.V. Ask students to examine the relationship between the weights and distances bycompleting a table. By using two students whose weights are considerably different,a pattern can be discovered. The results in the last column will beapproximately equal.Wei9ht of'Dista.nce wis froCI)) W+D W-D W+D \'vxDoel"son(w) blockDThe Genero.l Ru.le is : WI D... -- =--.W l D.VI. Students can apply the general rule to solve problems: For example, John has amass of 40 kg and stands 1.5 metres from the block. Tim balances the plank bystanding 2 metres from the block. How much does Tim weigh?"Give me a place to stand, and I will move the Earth."This is what the famous Greek scientist Archimedes (287­212 B.C.) was supposed to have boasted after discoveringthe law of the lever: WI x D I= w 2x D 2. Assume thatArchimedes weighs 150 lbs., and the fulcrum of the leveris 4,000 miles from the Earth. How far from the fulcrumwould he have to stand in order to move the Earth? TheEarth weighs 13,176 ,000, 000, 000, 000, 000., 000 ,000 lbs.339

Have a student bring a 5 or 10 speed bicycle to class.that the gears can be shifted. Put a piece of tape onHave the students count the teeth inEeach gear and record in Tanumber is not standard. Tvary from 52 to 39 teeth agears from 34 to 14.)Write the gear ratios andRecord in Table 2.The following activities afor student investigation:1. Select a simplegear ratio, for example,13 to 4, and set thegears to correspond.Check the gear ratio byslowly turning thepedals. The pedalsshould turn four timesand the wheel thirteen.(Hold the rear tirelightly to aid incounting the turns ofthe wheel.) Checksome 'other gear ratiosby counting pedal andrear whee] turns.~ble 1. (The -BTurn the bike upsidethe rear wheel of thehe front gears -cnd the rear-DAX_-~,'I=:':simplify.Rre suggested dGeo..r-XyTABLE 12. Select a back gear and use the smallshift to the large front gear. Continuechange do you notice in the back wheel?gear ratios?Nu.mb~of teet'"down so (bicycle.TABLE 2Geo-y Ro.tiO of Simplifiednu.m b el"R.o..tio of teeth Teeth Ro..+ioXtoAXtoBAXtoCB)(to Dc Xto h:DytoAEVto BVtc Cyto Dyto Efront gear. Turn the pedals slowly andturning the pedals at the same rate. WhatCan you explain? What are the corresponding3. Move the gearshifts so the chain is on the smallest back and front gears. Turnthe pedals at a constant rate. Shift only the back gear so that the chain travelsfrom the smallest to the largest gear wheel. What change occurs in the back wheel?Can you explain? What are the corresponding gear ratios?4. If the pedals were turned at a constant rate, which ratio would cause the backwheel to turn the fastest? Order the simplified gear ratios from largest to smallest.Students could use a 'calculator to change each ratio to a decimal and then order thedecimals.5. In riding the bicycle, which gear setting is the easiest to pedal? the mostdifficult? Experiment on the playground. Which gear setting allows you to travelthe farthest for one turn of the pedal? Devise a method for checking your prediction.6. Select a gear setting. Suppose you pedal at a constant rate (one turn per i(.second, thirty turns per minute, etc.). How far would you travel in 20 minutes?~3407. Select a gear setting. How many turns of the pedal are needed for the biketo t.ravel a distance of one mile?

341

(Materials Needed:An inclined plane (a board about two metres long), metre stick,several blocks to hold up the inclined plane, spring scale, cart(roller skate or toy truck)An inclined plane is a simple machine.It helps to make work easier.The workerpicking the boxup exerts a largeeffort but througha short distance.A) Block up one end of the boardto make an inclined plane.Measure and record the heightand length of the inclinedplane.This worker is sliding the crate upthe inclined plane and is pushingwith a smaller effort but over alonger distance.(B) Height cmC) Length cmD) Express the following as a ratio and its equivalent decimal.length of inclined plane7 h :': e '""iC'g'"'h:':t'-'o"'f;'-:l'"n"'c"l:;::l7.n=e~d.J:.p:;;:l::::a:'::n"'-e = =E) Hang your cart from the spring scaleand record its mass below.Mass of cartgramsIDEA FROM: Ratio and Proportion RevisitedPermission to use granted by Oakland County Mathematics Project(342

CO\\lTlI\lUED)F) As the cart is pulled up theinclined plane at a steadyspeed read the spring scale.What is the scale readingwhen the cart is pulled?gramsG) Express the following as a ratio and an equivalent decimal.mass of cart~7--"~P'7'-::--_---,-------:-;----;---:;-;---;----:;--=pull needed to move cart up the inclined planeH) How do the ratios in parts D and G compare?A second trial should be done changing the height and either using a new cart orthe same cart with some additional mass.IDEA FROM:Ratio and Proportion RevisitedPermission to use granted by Oakland County Mathematics Project343

BEING \tJTHE: MCDOL.E:Of: THINGS (ICut several differently shapedtriangles from heavyweightpaper. They should be about20 cm at the widest spot.1) Let the triangle swingfreely from the pin.2) Mark the point wherethe weighted stringcrosses the side ofthe triangle. Sketchthe line representingthe hanging thread.../,.3) Do the same for theother two vertices.Try balancing the figure at the point whereall lines intersect. Use the blunt end of (your pencil to balance on.THIS POINT IS THE. BALANCE. POINT OR CeNTER OF GRA\JITY.7) Find the meaning of the following terms.a) Angle bisectors of a triangle.b) Altitudes of a triangle.c) Medians of a triangle.Which of these will help you find a balance point of a triangle?II Cut a rectangle and a circle from the heavy paper. Using a plumb line, find thebalance point of the figures.Describe where the 'center of gravity is.(344

Materials Needed: One Celsius thermometer, one Fahrenheit thermometer, ice cubes,hot water and saltMeasure and record the temperatureof each item with the Celsius andFahrenheit scales.HOW CAIJ IT B" SOr Tf-IOUGI-\T WATERfROZE AI 32." ?HOT,C£:LSIUSPlot these pairs of temperatures on the grid below.Place the Celsius readings on the horizontal axisand the Fahrenheit readings on the vertical.a smooth path through the points.Using the graph determine -210,the temperatures in Celsius.I-180A) Normal body temperatures°c-----B) A warm day __C) A cold day __D) Water freezesIIFI50HIW,°c :I 1'.1.0:2,°c ~II 190LE°c I-GODrawgO-30 ·2.0 '10 0 10 2.0 30 'l0 50 GO 70 80 90 1001£~SfUTI I345

Materials Needed:REW.ii~W~EXPEj;2IENCEIce cubes, warm water, Celsius thermometer, stirrer, watch and aglass or cup(I 1) Fill the glass about 2) Add a couple of1"2 full of water.ice cubes andstart stirring.Read and record theItemperature every "2 4) If the ice cubesminute ..melt too fast addsome more.In the chart below record the startingtemperature and the new temperaturesevery1minute after you begin stirring.5) When the temperature hasn'tchanged in four readings-­stop.LI;:NGTH OFTIMESTIRRED Ill! MINUTESTEMPI;:RATURE: It-.)DEG.RE.ES CE.L~\USSTARTI1 1.1- ?. 2~ 3 3~ t+ 4.1- 5 5'~2: 2- 2 2:\(IIa graph like the one to the left, plotting the information you gathered inDRAW THE STRAIGHTLINE. WHICH BESTFi,S YOllRPOIl\lTSi I l~ 2. 2i 3 9i 4- l.f~ 5 5~ GTIME IN MINUTESIII You be the scientist.part I.How much did the temperature dropin the first minute?in two minutes?in five minutes?Let the horizontal axis be time andthe vertical axis be temperature as shown.in thirty minutes?Can you explain why the water and ice cube mixture didn'tget colder than O°C? (346

The distance a person can see on theearth is limited because of the earth'scurved surface.The formula D = 3.57 x jilallows usto determine the distance (D) we can seein any direction from a height (h). Thedistance, D, is in kilometres and theheight, h, is in metres.I CAt-l SEE ALL TilEWAY TO Tilt: J'--./'EDGE...~::.: rV~' ',....:_. .'.' ',.", :,:,.".,::,:·1·.· .. ·· .....•..•.:: '1 IIEarly sailors climbed into their crows nestshigh atop a mast in order to see greater distances.DOME 01"NOTREDAME-- .... DrDJOSERSTEPPYRAMID:.:._ ..... ,.·f:~~i~= PEAK.

The following activities investigate the relationship between volume and displacement.(Materials Needed:Two identical graduated cylinders, several objects having the samesize and shape but different masses (for example: two marbles,one glass and one steel)IIIb·.:-:....,.. .. 7.7.ct.2'" .'.. . 1517:--,--,~+20'. 1.6. ,,-A..SS It) STEEL _. 10GL .' '.O/'~1

In 250 B.C. Archimedes knew that an equal volume of gold and water did not havethe same mass. In fact, it was discovered when comparing equal volumes the gold wasalways 19.3 times as heavy as water.This ratio of masses is known as specific gravity (sp. gr.). Every substancethat sinks in water has a specific gravity greater than 1.00. For example.sp. gr. (brass) =mass of brassmass of an equalvolume of water= 8.44COMPARII\lG TflESE EQUAL VOLUMES OFWATER At-lD BRA~'5 1 KNOW THE BRA~5WILL BE 8A't T:E'5 AS HEAV~(BRASS 0 [WATER 0 ~Materials Needed:Spring scale, container partly filled with water, a variety ofobjects to be submerged such as small pieces of iron, brass,rocks, etc.Using the spring scale record each of the following.I II III Loss of mass in waterMass of Mass of (difference between I0 0the object object in and II) . g~'-'-'in air.[j1 t •.•. :?'gwater.nI]gIVspecific gravitymass of object in air= loss of mass in water ---Check your values with your teacher to see how accurate you are.Find the specific gravity of several other objects.349

~~lt~~ ~NDHALF-LIFE/,Materials Needed:Graph paper, set of 20 cubes with one face of each cube marked witha D /"""' ._,..------..,Each die in theset representsA particle decays, loses its radioactivity,when the side marked Da radioactiveturns up. It is removed from theparticle.pile before the next roll.IIIBegin by rolling all 20 particles.Remove any particles that decay (theside marked D turns up).Recordyour results in the table at theright.Continue rolling all active particlesuntil all of them have decayed.ROLL PARTICLE.'::>DE.CAYEQPARTICLE:"'"RE.MAIl\lIl\lG"'TART 0 2.0ArTER THE1rtR'.'>T R.OLL>ONLY ROLL THOe.E: 2-PARTICl!;'::> Tj 3ARE STILLRADIOAC\IVE. LJ-Do you think the results would 6be exactly the same if yourolled the 20 radioactiveparticles again?~5789(III Make at least five more trials, each time starting with 20 radioactive particles.As in part I roll all radioactive particles until they have all decayed. Recordyour results below.12.34-5G7

~~Jt~y RNDHALF-LIFE.(CONTltJUED)IVV~;;

QJ~t~Ur! RNDHALF-LIFE (Nuclear reactors are becoming more common because of the energy crises.again reduced by onehalf,are 1000that is half of1000 RADIO·ACTI\i£R'\RTICU:S.t1000I­ u~~ 500 . __ ~.. ,0,TI-1E TIME REQUIREDFOR I-1ALF Tf1E.PARTICLE':> 'TO DE.CAY I~REtERRED 'TO AS 'THE.HALF· UFE.~ 250 i

The half-life of a radioactive substance is the time it takes for half of the radioactiveparticles to decay.DURING THE:FIRST HALf,LIF'E 5GOPART!CLIO:SDECAY.t or ORIGINAL LEFTDURING THESECOND HALF­LIFE HALl" OF!!-IE REMAIN­!Me. PARTICl£SOCCAY

Albert A. Michelson (1852 - 1931), America's first Nobel Prize winner in science,made hundreds of measurements of the speed of light. His efforts to establish itsspeed covered a period of 53 years. He's most famous method is shown below.(Ot.-\Al-JTOtJIO35,38G METRESMichelson directed a strong beam oflight onto one face of an octogonalmirror. The light was reflected toanother mirror more than 35 kilometres distant on top of Mt. San Antonio. A sphericalmirror there reflected it back. At the exact moment the light returned the eight-1sided mirror had completed 8 of a revolution. The returning light was reflected intoa receiving unit.(1) What is the total distance in metres the light travels from the time it leaves thelight source until it returns to the receiving unit? __~~~ metresOver many hundreds of trials Michelson's average value for the speed of light was299,796 km/s.Using this value how far would light travel in the following lengths of time?GET READY,GET SET,2) Time 2 sec 5 sec 10 sec 1 min 8 minDistance 1/!light hastraveledinkilometres.(

fH;-7~J##A/Hrp£fg-:Z5V£.§M-Z-E.7 (CO"lTi~u£.\"))3) The distance between the earth and Alpha Centauri, the nearest star outside ofour solar system, is 4.3 light years.A light year is the distance light can travel in one year.Using the approximation of 300,000 km/s for the speed of light find how farlight would travel during the following time intervals.1 minute km1 hour ____________ km1 day km1 year km4) How long would it take for light totravel the following distances.SAIJ•FRAIIlCIS 0A) From Mexico to Canada?B) From New York to San Francisco?____ ssC) Around the world? (40,200 kilometres)sD) From the earth to the moon? (386,000 km)E) From the earth to the sun? (150,000,000 km)ssEstimating the distance from a lightning strike.Sound travels much slower than light. In fact, it takes about 3 seconds forsound to travel one kilometre. Knowing this a person may calculate how far he isaway from a lightning strike.Follow these directions.A) Count the number of seconds between Seeingthe lightning and hearing the thunder.B) Divide the number of seconds by three.This will approximate how many kilometresare between you and the lightning.5) Complete this table of values.Seconds between lightningflash and thunder sound3 6 1 5 9 10Distance inkilometres355

RtFLECTIONS2~OITJ3J13R(Two figures can alsoother figure when ithave a line of symmetry--if oneis "flipped" across the line ofIII,I,,I,,III,II,If the dogs exactly fit when foldedover the line, one is called areflection of the other. The lineof symmetry is also called a lineof reflection.II,• I,IIIIIII,figure exactlysymmetry.~--l1',1:\ "::..: "I "I.·..I' •,: :,"II r "I: .•.".: , , , , , , , ,fits onto theooJ'1------,Imagine the word "dog llbeint; llflipped"across the dotted line. Would itexactly fit the other "dog"? Try itby folding the paper, using a piece of:Jlastic, or using a mirror.(for each pair of figures decide if one is a reflection of the other.piece of plastic or a mirror, or fold the paper to help you decide.You may use a356~-------------N(IDEA FROM: Mira Math for Elementary SchoolPermission to use granted by Creative Publications, Inc.

®RtFlECTIONS(CONT)0,0,®D®®2~OITJ3J11~(TV10J),\G\\, ,\\\,G)\\\\\\,\\@ M: MI,AlA ,TiTHiH,, II-j@ ,bid ibid,I@IDEA FROM:Mira Math for Elementary SchoolPermission to use granted by Creative Publications, Inc.@ MAIAYAIAMMAIAYAIAM357

MIRROR, MIPPOQON TI-I[; WHLL(Materials Needed:4 straight pins, mirror, cardboard of softwood to fit under paper, protractor, blockof wood for support1)ITape the mirrorto the wood andmake an arrangementlike indrawing. Besure c is per­BLOCK. or WOOD rOR SUPPORT-.>: : ";." -.", ..;.:.-. ", .:. -":"'::.,.; ", ~"': •._"," _.. :...·.'·.c': :':':.::,: . ~'.:.' '. ".}',: .',.,":"::: .....:.;..::A PIECE OFCARD80ARD OR50nWOOD UtJDIOR­'. NEAtH THE PAPERPROtECTS TI-l£ DESK.' TOP AtJD MAK.E:-:' IT:' EASlER TO PUSH:: \1-> THE PINS.pendicular to a.IIDraw in line segment bIIIPlace two pins, #1 andto represent an incoming#2, upright along lineray of light.Any anglesegment b.They shouldwill work.be about 5 cm apart.2) Stick pins 113 and #4 in the paper so that when you lookinto the mirror these pins will lie in a straight linewith the reflection of pin III and 112. Moving your headback and. forth will help you find this position.(3) 0. -j/-..:":- _bRemove the mirror and draw the line passing through thetwo new pin holes. Angle i is called the angle ofincidence and angle r is the reflected angle. Measureboth of them. -m. L i = 0 m L 'r = 04) Repeat the experiment with different angles, for the first use a larger incidentangle than the previous trial. For the second, use a smaller angle.Angle of incidenceAngle of reflectionTrial 111Trial 112Trial 1135) You be the scientist. Using the data above make a statement relating the measuresof the angle of incidence and the reflected angle.358(

Materials Needed: straight pins,small hand mirror, cardboard or softwood to fit under paper, ruler, blockof wood for supportI Getting set up.On a piece of paper backed withcardboard draw line t, set up themirror and place the object pinabout 5 centimetres in front ofthe mirror as shown in the diagram.BLocK orWOOD toRSUPPOR.T~I HI£. Sl\ORT£.STOBJECT f DISTANCE. FROM THE=======P=II\l==-=t0BJ£.CT ToTHE:MIRROR IS T\4EOBJECT DISTAtIlCE.IISight lines.Anyone who has ever played with a toy gun knows that to sight in a target the rearsight, front sight and target must be in a straight line. We will use this samemethod to sight in the image of the pin.RE:I'LECTIOIJ OF" OBJECT PINPlace pin #1 about 2 cm in front of the mirrorand to the left of the object pin. With youreye near the table level place pin #2 so thatit is in a straight line with pin #1 and thereflection of the object pin as shown in thediagram. This is a sight line.Using the same procedure, line up two morepins along another sight line from the righthand side.III Finding the image position.Remove the mirror and its support.With a straightedge draw the linethrough pins #1 and #2. The linemust extend behind line t sincethat is where the image willappear to be found. Do the samethrough pin #3 and #4. The point of intersection is the image position.359

IV~~~~~~( CONl"lNUED)Making some measurements and drawing a conclusion.Measure the image distance and the object distance to the nearest one-halfcentimetre.A) Object distance cmB) Image distance cmDo two additional trials. For each trial use a new object distance. Enteryour new data into the table below.(Object distanceImage distanceTrial 1/1Trial 1/2What relationship do you see between the object distance and image distance?MIRROR, MIRRORON THE VJALL, '1JVJI-\OSE THE ~FA1RE5, OF'I-\£.I"\ All?If the wicked witch is standing three andone-half metres in front of the mirror,how far from her does her image appear?(VMirror image.Trace the line land the scalene triangle onto a clean sheet of paper.I PLACEMIRROR I-\EREIn the same manner as above find the image points of A, Band C. Connect theimage points to form the image of the triangle.Measure the lengths of the sides of the original triangle ABC and the mirrorimage of the triangle.How do they compare?(360

MllUllllllPlllE[J~EFl E[ 1111[]IINIIS,Materials Needed:Two mirrors, two wooden blocks for support, cardboard forprotection, protractorIOn a sheet of paper makea 90° angle.equipment as shown.Arrange theCARDBOARDBACKH.lG UI\lDI:RPAPER.PLACE AI\lDeJECT, SOMEWHATOFF CENTE'R, BETWE£I-:lTHE TWO MIRRORS ..... " ....... , .. ,.. , ...,..Look into the mirrortable level.many images docount?II What is the effect of varying the angle between the reflecting surfaces of the mirror?1) Begin with the mirrorsat a straight angle of180°. Enter the datainto the table at theright.2) Do the same thing onlyuse the angles in thetable. Write yourresults in the table.Number ofImagesNumber of Plus Angle MultiplyImages Object (N) (A) N • A180°120°90°72°3) Look for number patterns in the data table. What patterns did you find?III Using the patterns in the table and the two mirrors find the angle needed toproduce the number of images indicated in the table.Number of ImagesAngle Between Mirrors5678910IDEA FROM: Laboratory Investigations In PhysicsPermission to use granted by Silver Burdett General Learning Corporation361

(Materials Needed:Magnifying mirror, candle, 4 cm x 4 cm pieces of glass and clayA concave mirror is needed.An ordinary magnifying cosmetic or shaving mirrorwhich can be purchased at most drug stores is of this type.I1)Find the focal lengththe mirror.eRO",,,, e.E:C'IONOF A 5P14ERICI'.LMIRROR."-------0"" e ........,If the sunis out you can reflectthe sun's rays to a finepoint on a piece ofpaper. Use caution~ Ifo , ,~.,. \0\ \,Getting the feeling. c)\c o\, ,, ,of\ ,'~~THI: POIMT WHERI:ALL THE SUf\lSRAYS AREFOCU5ED ISTHI: FOCAL POlfJT. _THE DI'.:>TANCEBETWE.EN TI-IlSPOIIJT Af\lD THEMIRROR IS THEFOCAL U:IJ(;TH.you have a good mirror it will have the same effect as a magnifying glass andmay give a very hot focal point.If the sun is not out use any object that isa long way off and project its image onto a piece of paper.iscm.The focal length(2) Describe images. Put a candle in front ofa concave mirror as shown. Using a pieceof paper as a screen find the image. Describethe image using general terms (bigger,smaller, same shape, etc.)Move the candle about 2 metres in front ofthe mirror. Find the image with the paper.Again describe the image in general terms.IMAGE OF CAWOLE FLI\IJlCOlJ A PIECE OF PAPERTHE OBJE.GT MU~T BEAi A Dl'.:>T/llJCE FROM THEMIRROR GREATER THANTHE FOCAL LENGTH.(362

IISome exact measurements.Geometric shapes marked on clear material (glass or plastic about 4 cm x 4 cm)make nice shapes to project. A flashlight can act as a light source. The4 cm x 4 cm glass can be mounted on a piece of clay.,,:......... '/\" ~~.'.", "AD OBJECT". ~' MAl

(To project an image using a concave mirror theobject must be a distance from the mirror greaterthan the focal length. Images that can be projectedon a screen are called real images.I Placement of the object in the followingregions provides for the best student investigations.A) The object placed between the focalpoint F and the center ofcurvature C. The image willappear beyond C, inverted,be reversed right to leftand be real.CoC>~ IML\GETHE C(!JTER 01'" CURVATURE,C, IS TWICE AS PAR FROMTI-lE MIRROR AS THE mCAL:LE"lCTH, F.This is used in movie projectors.(B) The object placed at the center of curvature, C.The image will also appear at the center of curvature.reversed right to left, the same size and real.It will be inverted,REfli-CTINC TELESCOPEIMAGELIGHT FROMDISTANT STAR,OBJEC.T.C) The object placed beyond C. Theimage will appear between F and C, beinverted, smaller, reversed right to leftand real.reflecting telescope.This is the principle behind aThe largest reflectingtelescope in the U.S., having a diameter of 200 inches,is the Hale telescope on Mt. Palomar in California.IIImage distances can be calculated.located from the364 measurements... .­-:"(1'f ,,,,For any concave mirror the image distance, i, can becalculated if the focal length, f, and the objectdistance, 0, are known.III\:1 Using the formula - + - = - or the equivalent formula'~'o i f. o· f>11. = will determine the distance the image iso - fspherical mirror. Students can check their calculation by actual(

BENDING OF LIGHTk7[~(7?7CrIaNLight travels at different speeds through different materials.from air into water at an angle it changes direction, we say, it refracts.When light goesWith this activity you will find a number called the index of refraction that hasto do with the speed of light.Materials Needed:4 pins, ruler, compass, thick piece of glassStep A 1) Make line labout 10 cm long.2) Line p isperpendicular to line l.Ip,3) Angle i, the incident"1,,.angle, should be aboutI l,"'- "2 45°.,I,II, about 5 cm apart.I 4) Place pins III & 1/2 in line IIStep B~ the glassplate flat sothe top edgelies on l.4) Place pins #3 & #4so they lie in astraight line withpins #1 & #2 thatyou see through theglass edge.pi ,, .:-.\I 2) Trace along the bottom of the"1 glass plate and label it n., 3) Looking from the table sightpins #1 & #2 through the edgeof the glass plate. Move yourhead until they are in astraight line.Important: Be sure to look through the~b"'f"/=~========~ glass, not over the top.Step C 1) Remove the glass plate.2) With a straightedgedraw the line throughthe pins #3 & #4 upto line n.01I3) Connect the pointsof intersection onlines l & n.This shows how the light rayas it goes through the glass.refracts365

Step DBENDING OF LIGHT?![r(7?7CrIGNpr-: ,III(COtuTllJUE:D), III: p,190 0, , .;-b,.,, ,1) With a compass make thelargest circle you canwithout crossing line n.2) Draw in line segments R andT so they are perpendicularto p.(9d'1TFind index of refraction.Measure Rand T to the nearest millimetre.of refraction.Lengths:Express to two decimal places.R= nun~--T = ____ nunThe ratio of R:T will be the indexRatio:R- =T(Index ofrefraction)(Try this activity two more times using different incident angles--one about 30°and the other about 55°.Determine the index of refraction in each case.Take an average of the three ratios for your final value.'l."'TABOUT "-15°If..lCIOEtJT AlJGLE L (1\\30VC) RAllOABOUT 30°RATIOABOUT 55°RATIOLEl\lGTI-\0, RLE.NGTI-\ O. TThe average of the three ratios is(366

BENDING OF LIGHl=(7[(?1?l[rIL7NWhen light passes from one tranSparentmaterial into another its speed changes.If the light hits obliquely this change ofspeed will be seen as a bend in the lightray at the boundary. This phenomena isknown as refraction.i IS T~E I\.lCIDEfJT ANGLE, MEA'SUREDBETWEEN T\.it:. IMCIDEl\IT RAY AND THe.PE:RPEIUOICUl.AR AI Tf1E POll\lT WHERE'THE IIUCIDEl\lT RPN It-lTERSECTS THE:.GLA'?,S.Several classroom demonstrations of refraction aresuggested below.I Put a pencil obliquely into a glass of water. Therefraction of light at the boundary of the air andwater will give the illusion that the pencil isbroken.IIPlace a coin on the bottomof a metal pan. Have studentsback up so the lipof the pan just hides theCOlt-!coin. With students stayingexactly the same position slowly add water. As the water level rises thecoin will become visible.III Give students a glass prism or rectangular piece of plastic or glass and havethem place it on a picture. As they tip the prism forward and back the pictureseems to move under the glass.The ratio of the speed of light in a vacuum to the speed of light in any otherparticular transparent material will give a constant value.of refraction (N).speed of light in a vacuumspeed of light in water1. 33 = N (water)speed of light in a vacuumspeed f I ' h 'd' d = 2.42 = N (diamond)0 ~g t ~n ~amonThe index of refraction of air is 1.00It is called the index367

BENDING OF LIGHT?![(?!?{CrIL1N(COIVTIIVUED)(pages.A geometrical method of finding the index of refraction is shown on the studentAn alternative method will be shown here.Willebrord Snell, a Dutch mathematician,studied this phenomenon and stated the followinglaw.Snell's law:N(air) xSince N(air)N(glass)sin i = N(glass) x sin r1.00 this can be stated(1. 00) sin isin rAn example of how this is used is shown below.Step A through Con the student pagesoutline the procedureto set up the equipmentand find anglesi and r.LAIR//_.~/;;;J/j'I//J/GLASS///i-r / MLr- 2.7/ / / IIIi / / 1// ~N(glass)N(glass) = 1.551.00 sin i \sin r= (1.00)(.707)(.456)(Students will need a table of sine values to use this method.368

C£OJSARYbarometer. An instrument for measuringthe pressure of the atmosphere.a. aneroid barometer. A barometermade from a partly evacuated, airtightmetal box which is hooked to ascale.b. mereury barometer. A barometermade from a mercury column supportedabove a bowl of mercury.eenter of eurvature. A sphericalmirror is thought of as part of asphere. The center of the sphere isthe center of curvature of the mirror.eenter of gravity. The points whereall mass or weight of a body seemsto be located. Also called balancepoint.eoneave mirror. A curved mirror suchthat two rays from a light sourcewill intersect. The reflecting surfaceof the mirror appears to beII caved in. tIfoeal length. One-half the radius ofa spherical mirror.foeal point. When parallel light raysfallon a curved mirror, the pointwhere the reflected rays intersect isthe focal point.fulerum.Support for a lever.fulcrumgravity. The mutual force of attractionbetween bodies. A measure of thegravity is weight.half-life. The length of time requiredfor half of the radioactive particlesin a substance to decay.image length. The measure of the distancebetween a mirror or lens and theimage formed by the mirror or lens.ineidenee, angle of (i). The anglebetween the incident ray and the perpendicularat the point of incidence.I,,,i: PERPEJJDICULARindex of refraetion. The ratio of thespeed of light in a vacuum to thespeed of light in any transparentmateriaLlever. A rigid bar which can rotateover a fulcrum.light year. The distance light cantravel in one year, ~ 9.5 x 10 12 km.mass. A measure of the amount of matterin a body. In the metric system thisis expressed in grams or kilograms.objeet length. The measure of the distancebetween a mirror or lens and theobject in front of the mirror or lens.pendulum. A suspended mass which isfree to swing back and forth.foree. A push or pull tending to produceor prevent motion. In the metricsystem the unit is newtons or dynes.pressure, atmospherie. The force perunit area caused by air. In the metricsystem this is measured in newtons/cm 2or dynes/cm 2 • In the English systemthis is measured in pounds/i~2.369bar-pre

PHYSICS GLOSSARYref-weireflection, angZe of. The anglebetween che reflected ray and theperpendicular at the point of reflection.: PERPEMD1CULAR,~Lf:CTf:DIIRA.Y(refraction. The bending of light asit passes, at an angle, from onetransparent material to another.specific gravity. The ratio of themass of a given sample to the massof an equal volume of water.weight. A measure of the force ofgravitational attraction between twobodies.(("370

:to MATHEMATICS AND SPORTSA 22-pound salmon caught on a 16-pound test line--lOO metres in 10.4seconds--lst down and 10 yards to go-­bottom of the 8th inning, score tied4 to 4, runner on 3rd base, 2 outs, a3-2 count on the batter--Evert leading30-1ove in the 1st game after winningthe 1st set by a score of 6-3--Anthonyneeds 2 strikes in the 10th frame tobowl a score over 230--and the numbersgo on and on. There is hardly a sport,from a simple game of basketball betweentwo friends to the complex operationsof the Olympic Games, that doesnot incorporate numbers, arithmetic orsome type of measurement.Most students are interested in physical activity to some degree--from ridingbikes to playing competitive sports. This interest can be used to motivate students,and sports can become a way of showing applications of mathematics in familiar situations.Being able to relate to your students by knowing who is on the girls'volleyball team, who participated in the Bike-a-thon against cancer or who owns ahorse and takes riding lessons may create a better atmosphere in the classroom.In this section on MATHEMATICS AND SPORTS it is not necessary for students orteachers to have an extensive background of terminology and rules relating to sports.Many activities explore the "common, everyday" uses of mathematics for recordingstatistics and scores. Each classroom probably has a student "expert" who can explainthe terminology and rules.The student pages are roughly grouped into three parts. The first part includespages that provide or have students investigate general information about sports.In studying a Soccer Ball students investigate ideas about the vertices, edges andfaces of polyhedra.The second part includes several activities that students can use to determinehow physically fit they are. The activity Are You Physically Fit? provides the standardsby which a student can become a member of the Presidential Physical Fitness team.371

INTRODUCTIONMATHEMATICS AND SPORTSIndividual sports and the mathematics involved provide the ideas for the page~l\in the third part.Track and field, bowling, baseball, football, ski jumping,shuffleboard, boxing, tennis, cycling, and boating are all used to cover such diversemathematical topics as computing range, mean, median and mode;interpreting and makingscale drawings; ordering fractions; and measuring distances and times.simulations of athletic events, e.g., Bowling with Dice, Fraction Bar Football,Classroom Decathlonare included among these pages.SeveralSeveral of the pages make use of records. As these records are broken, you cansubstitute current records, especially metric ones as the transition to the metricsystem in the United States is completed.Pages in MATHEMATICS AND SPORTS can serve as a springboard for other areas ofstudy. Sports, especially with the Olympic Games, has an international flavor. TheLittle League World Series involves teams from other countries, and high school andcollege wrestling and gymnastics teams are involved in exchange programs with teamsfrom foreign nations.these countries.Use of these events can interest students in studying aboutThe concept of consumerism could be explored by examining costs ofequipment, money spent on concessions, salaries of professional athletes, admission/prices to sporting events, attendance at these events, and capacities of the arenas\,or stadiums that hold these events.In conclusionl;lHdon't strike out"; "be on the ball"; "get out of the startingblocks quickly"; and use some pages from MATHEMATICS AND SPORTS to capture the interestof your students and provide motivation for the study of mathematics.and372

Use Newspapers and MagazinesSave clippings from the newspapers and magazines that include sports records,statistics about players, advertisements about equipment and events, and picturesfor the bulletin board. The sports section of a newspaper is the easiest place tofind facts and figures, but you might also find sports information in magazines andother periodicals. Showing the relationship of sports to other problems in societycan be interesting. See the graph below. Questions could be made from the chart foruse by you or your students.' .." ~, . '.' ..' .. . .- .w UGO; . " -i e~'. ",", .... : '".',.' .. . .:: .: .'.:~ :'.::'," ". '. ': .. : ','":'.. : ... ' - .. " ." ,",': ., .. ' . .:.,: ... ... . "',', : ", .:.': ':',. :.:"," ""... " ..',' ".'," .10 2.0 '30POLLUTANT LEVEL'5 (PARTS PER \-lUIJDRE:D MILLION)This chart shows the effects of certain air pollutants on the performance of crosscountrytrack teams as pollution in an area became worse. Pollution levels wererecorded one hour before each meet.373

World swim championships(Men(PRELIMINARIES)20ll-meters BackstrokeFIRST HEAT - 1, Lutz Wan la, East GermanY,2:11.62, 2, Bob Tierney,. United States, 2:12.77.3,Reinhold Becker, West Germany, 2:15.47. 4, GaryAbrahams, Great Britaln, 2: 15.62. 5, Igor Orner·.chenko, Sovjet Union, 2: 16.80. 6, Lorenzo de faTorte, Colombia, 2: 29.60.SECOND HEAT ~ 1, Poul Hove, United Stales,2:08.73, 2, Abdul Karin RessonO, The Netherlands2:09.97.3, Steve ,Pickell, Canoda, 2:11.20. 4; Krasimir Pefkov, Bulgoria, '2:11.79.5, Canrado Porto,Argentina, '2:12.78. 6, EnriQue LedesmQ,' Ecuador,2:22.90.THIRD HEAT -,', Roland Matthes, East Germany,2:08.73. 2, Sollon RUdolf, Hungary, 2:08.92.3, James' Carler, Great Britain, 2:09.77. 4, Fron_eisea Santos, Spain, 2: 11.94. 5, Lapo Cianchi,2: 13.80.FOURTH HEAT - 1, Mark TonelH, Australia,2:06.08.,2, Zoltan Verraszto, Hungary, '2:07.91. 3,Santiago Esteva, Spain. 2:08.94. 4, Butch Batchelor,Canada, 2:13.12. 5, Bado Schlag, West Germa_ny. 2:16.48. 6, Gener otero, Colombia, 2:19.02. 7.Jose Luis Lopez. Ecuador, 2:28.15.40ll-meter FreestyleFIRST HEAT - 1, Gordon Downie" Great Britain,4:05.60. 2, Peter PeUersson, Sweden, 1:06.22.3, Alexondr Somsonov, Sovief Unian, 4:1 .67. 4,Michoel Ker, Con'ado, 4: 13.7Q. 5, Juan AI/redouribe, Colombia, 4:3L5{>.SECOND HEAT - 1, Frank Pfulze, East Gl,'many, 4:04.25. 2, Rainer Srrahback; East GermanY,4:05.02. 3, Benot Gingsio, Sweden, 4:09.72. 4,Jim Green, CanQdo, 4: 15;47. 5, Krosimir Enchev,Bulgaria, 4~20.27. 6, Gullermo' Pacheco, Peru,4:20,99.THIRD HE'AT - 1, Bruce FurnIss, UnitedStotes, 4:03.31. 2, Werner Lope, west Germany.4:07.37. 3,Max ,MetZker, Australia; 4:08.53. 4,Breit Taylor. New Zealand, 4:09.24.5, Hel'\k Elzer~man, The Netherlands, 4:12.52. 6. Tomas' Becerra,Colombia, 4:22.00.FOURTH HEAT - 1, Tim Shaw, United States..4:04.2:1. 2. Grpham Wlndeatt, Austral/a, 4:04.76. 3,Andrei Kr.ylov, Soviet Union, 4:05.6,2.4, Marc L'aIzard,France,!XJurg,4;28.91.4:07.70.'5, 'Josy Wllwerf, Luxem'Women(PRELIMf~ARIES)lOll-meter ButterflyFIRST HEAT ""',1, Jill Symons, United states,1:03.56. 2, Gunl!lo Andersson. Sweden. 1:05.28. 3,Natalia Popova, Soviet Vnlon, 1:05.41. 4. FCQvlaNadalutll, Romania, 1:05.90. 5; Evp Lyndonyl,Hungary, 1:06.50•. 6, Montserrat Mojo, Spain,1;01.41.SECOND HEAT T 1,' Camlf!~ Wright, UnitedStoles, 1:03.74.:l, Marlo de Milagros Paris, CostaRico, '1:04.48. 3, :ramara Shetofostov6, SovietUnion, 1:04.97: 4, Kuniko Ba'nno. Japan, 1:05.09. 5;RosemarIe Ribeiro, 1:,05.41. 6, Schiavon DonateHa,It.aly, '1:06.15.' 7. Beate .Jas.ch, Wi/st Germany,1:07.09. 8, Marla Teresq Carrero, EcuQ{jor;1: J4.05.THIRD HE",., - 1. Rasemarle 'Kother, EastGermany, 1:02.83. 2, Gudrun Beman, West ,Germany,1:04.42. 3, LYnda Hanel. Australia, 1:04.62.4, Wendy Quirk, Canada, 1:04.73. 5, Lynne Rowe,New Zealand, 1:05..23. 6. Joanne Atkinson, GreatBritain; ·1:06.36. 7; Omla Rampazto.·ltaly, 1:06.41.8, Maria Matllde Menacal, COloO)bla, 1: 10.43.""FOURTH HEAT - 1, Kornella' Ender, EostGermany, .1':02.86. 2, Barbara Clar,k, Canada,1:04.65. 3, Nire Stove. Australia, 1:'04.85, 4, Mon·laue R9dah'l, New ZePredicting a Victory> and WeatherSome other examples of pages that involve the useWorld Track Records--Field Events>~ndWater Conditions.As much as possible relate any statistics to those of your students and/orschool records.374

Page 6C REGISTER-GUARD, Eugene, Ore., Monday, March 18,A half-rrillioJ:1 going cross countrySkiers turn from downhillPARK CITY, Utah (AP) - Skiers in increasing numbersare attempting to escape the expense and hassle of thedownhIll runs by turnIng to a different version of the sport ­skiing cross country_Although skiing cross country can be a lot more physicallydemanding than skiing downhill, its participants arefinding it cheaper, less competitive and something the wholefamily can enjoy together."Old people go at their own pace; young people becomeaware of nature," says G. Ingval HDewey" Tofson, who runsa ski touring shop at Park City. "You get away from the liftlines and the expenses."Tafsan's shop is among establishments across the countrythat report booming sales and rentals of equipment forskiing cross country, also sometimes called ski touring.MANY RESORTS full of chair lifts and gondolas fordownhIll skiers are now grooming trails and stocking equipmentfor the ski tour enthusiasts.A skiing magazine estimates there were 125,000 crosscountryskiers in the United States in 1970, and 500,000 thisyear.Cross-country skiing involves light, thin boots which areusually fastened to a lightweight wooden or fiberglass skionly at the toe. Then, with a pair of bamboo poles, and theright wax on the ski bottoms, a skier can easily make hisway up and down hills and across almost any terrain.It is very difficult to go uphIll with normal downhill skis.In the Lake Tahoe area on the California-Nevada border,there are now 13 cross·country schools, most of them'Old people go at their own pace;young people become aware of nature'set up in the last two years, says Skip Reedy, who runs theNordic Ski Center at the Squaw Valley ski resort."Cross country has been getting popular in the East andnow CalIfornia and the West is getting on to It," he said.TimberlIne Sports in Salt Lake City sold about 39 pairsof cross-country skis in the 1970-71 season, but 787 pairs lastseason, said Dave Smith, manager until recently.The article abave-shows trends, costs, problems, etc.article can be related to the situation in your town."THERE ARE a lot of people who downhill ski actIvely,but also do cross country because it lets you get out into themountains alone in winter", said Smith."It's a good family sport," he said. "In downhill, Dadtoes oIf one way and Mom another. But people of differentabilities can tour on the same terrain."Nonetheless'; downhill skiing has not lost its attractionsto its devotee. Skiers who have tried cross country stili talkenthusiastically about the thrills of speeding downhill. Thereare other advantages, such as lithe company of lovely women,"champion Stein Ericksen notes on the slopes and in thelodges."Downhill skiing is the immediate thrill; iCs very spon~taneOllS," says Joe Buys, a certified downhill instructor whonow manages a cross-country ski ·shop at Park City."Cross"country skiing will never come as far in popularityas alpine (downhill) skiing." said Ericksen, the winner offour Olympic skiing Gold Medals who now directs skiing atthe Park City resort. "Americans are not that dedicated tophysical conditioning."ERIKSEN SKIED cross country often during his youthin Norway. Cross~country skiing on a golf course where ParkCity offers many lessons is not much work, but touring inthe wilds can be strenous, he said.There are some suggestions. however, that conditioningmindedskiers might like ski touring because it is morestrenuous."The American people have been quite fitness~consciousthe last few years, and are finding winter weather can beenjoyed in activities similar to summer hiking," said SvenWiik, a former U.S. Olympic Nordic ski coach who now runsa ski lodge at Steamboat Springs, Colo.He agrees that ski touring "has been grOWing tremeD:'dously the last few years."1450me slope skiers are also trying ski touring, andwe're probably going to see more and more people who enjoyboth," he said.But for skiers who make a choice based on expense, skitouring probably would Win.While the boots, skis and poles for downhill skiing maycost $200 and ski lift tickets $6 to $12 a day, a cross-countr)package usually can be accquired for no more than $100.Parks, forests and golf courses can be used free in mostcases.The situation in theHave students investigatesales of sports clothing at local stores, possible needs for more tennis courts,bike paths, etc.Permission to use granted by The Associated Press375

Certainillustrationscan be used asthe focalpoint of abulletinboarddisplay.Permission to use granted by Old Oregon

(PAGE: 5)TENNIS BAllSYellow or whiteCan of 3reg. 2.99............. $1 99Collect ads to use in consumer~relatedactivities.percents of decrease.Have students find amounts andThey could compareprices of identical (or nearly the same)equipment at different stores.Build upon student interest to encourageprojects.were not a constraint?What criteria would you use in selecting an item of equipment if moneydetermine what you would buy and why.Select an item to research (baseball bat, tennis racquet) andIf money were a problem and you could onlyspend 75% of the cost of that "best" item, what criteria remains unchanged for youand what do you give up to save the money?;--_....--_..-----------GOLF CARTour reg, $17,99cart at the pay lessspecial reducedprice of .••;14 99GOLF BAGour reg. $19.97 quality golf bag atthe reduced price of ...h~Reg.16.9912 99FRESH WATERSPINNING ROD & REEL6 '7' 2·picco hollow gloH rod w'il06fcel ,t)ooled 1SO yOrd,. Bib.111(1"0, 2 "~lr

(PAGE

CONJEflTSMATHEMATICS AND SPORTSTITLEPAGETOPICMATHTYPESports and Numbers382Recognizing sportsnumbersDeveloping number senseWorksheetThinking AboutSportsNow is the TimeWhere's It At?383 Classifying sports Classifying sports usinga continuum384 Classifying sports Classifying sports usinga scale385 Locating sports facilities Using time to estimatedistancesTeacher ideaWorksheetWorksheetDropping the Ball387Determining reboundcapacityMaking and reading agraphActivity cardStudy a Soccer Ball388Investigating a soccerball.Investigating a polyhedronActivity cardSports Matohes389Recognizing sizes ofsports equipmentEstimating diameters andmassesWorksheetActivity cardVolume391Recognizing volumes ofsports equipmentCalculating radii and volumesWorksheetClassroom Decathlon392SimUlating a decathlonFind percentsCollecting and recording dataActivity cardTo Each His Own395Measuring personal dataand performancesFind ratioDrawing graphsActivity cardMuscle Fatigue396Determining physicalfitnessMaking and reading a graphTeacher ideaMy Heart Throbsfor You397Determining physicalfitnessComparing ratesActivity cardStep Right Up398Determining physicalfitnessComparing ratesReading tablesTeacher ideaActivity cardWhat's Your Type400Determining body typeUsing proportionsWorksheetCounting Calories401Determining calorie useMultiplying decimalsWorksheetAre You PhysicallyFit?402Determining physicalfitnessCollecting and analyzingdataTeacher ideaActivity card379

(\TITLEPAGETOPICMATHTYPEOlympics - 1976 406Analyzing Olympic resultsMultiplying and addingwhole numbersWorksheetGive Me a Head Start 407Using staggered startsfor a track meetFinding circumferencesWorksheetWorld Records 408Comparing world recordperformancesUsing proportionsWorksheetWorld Track Records:Field Events 409Comparing world recordperformancesAdding and subtractingfractionsWorksheetWorld Track Records:Running Events 410Comparing world recordperformancesAdding and multiplyingfrac"tionsWorksheetDecathlon Decoded 411Comparing decathlonperformancesAnalyzing dataWorksheetBowling With DiceScorekeeping inBowling412413Simulating a bOWlinggameScorekeeping instructionfor bOWlingAdding whole numbersAdding whole numbersActivity cardWorksheet(\Odds and Ends AboutBowling414Measurements involvedin bowlingMUltiplying and dividingdecimalsWorksheetTake Me Out to theBall Game415Measurements of abaseball fieldMaking a scale drawingWorksheetBase­Statistics:ball416Finding batting averagesand won/loss percentsDividing whole numbersWorksheetSlugger417Finding sluggingpercentageMUltiplying, adding, anddividing whole numbersWorksheetBaseball AutoRacing Records418Batting averages andcar speedsRounding decimalsWorksheetFootball Scores419Using league standingsFinding range, median,mode, and meanTeacher ideaWorksheetFraction Bar Football420Simulating a footballgameOrdering frac"tionsGame(380

TITLEPAGETOPICMATHTYPEFootball Injuries 421Locating footballinjuriesFinding percentsWorksheetFlight of Ole '98 422Simulating glidercontestMeasuring distance and timeTeacher pageShuffleboard 423Scoring shuffleboardAdding and subtrac"tingwhole numbersWorksheetPredicting a Victory 424Comparing measurementsbetween boxersAdding and subtractingmixed fractionsTeacher pageWorksheetFamous BlackAmericans Puzzle 425Identifying famousathletesMUltiplying whole numbersWorksheetPuzzleFamous AthletesPuzzle 426Identifying famousathletesAdding timeWorksheetPuzzleTennis, Anyone? 427Measurements of atennis courtFinding length, width,perimeter, and areaWorksheetGet in Gear 428Finding bicyclegear ra"tioSolving proportionsWeather and WaterConditions 429Recognizing favorableboating conditionsReading a tableFinding percentsWorkshee·tStudying Some Sports 430Analyzing attendancefiguresReading a tableWorksheetAnalyzing Activities 431Analyzing participationReading a tableWorksheet381

Each of the following numbers is associated with one or more sports. Place one ormOre of the letters in each blank to identify the associated sports. Beside each number,briefly explain what the number means to the sport you have chosen.(1) 714 @-2) 33) 9.0 a) auto racing4) 500 b) baseball5) .367 c) basketball6) 100 d) bowling7) 300 e) football8) 18 f) golf9) 5 g) soccer10)i ~9 h) tennise11) 15-30-40 i) track and field12) 1113) 6......_..--.-- ~List other numbers that are associated with one or more sports.mates can guess the sport and meaning you had in mind for each number.See if your classe382

1. The following activity deals with several ways of classifying sports. You can drawthe continuums on the overhead or chalkboard and have students place individual sportsat an appropriate position along the line.Several examples are shown.a) Amount of activity~ LEAST ACTIVE MODERAT£LV AC'TIVECHECKERSGOLFMOST ACTIVE ~SOCCER.b) Preparation needed~ 1- SPO\\I'TAE.OU~PL_A...;tJ~IJ~E:...;D......:~JOGGI\\IGHOCKEYc) Environment needed~ t-JATUR.ALFI5HHJe;.COl\J5TRUC'TEOTE:IJt-.lIS ~'\ d) Number of participantst:"LE:\tIBLE~ BIC'ICLII-lGLIMITED BY RULESBASKE:'TBALL ~2. Another way of rating these sports would be to assign each sport a value from 1through 10 with 1 being at one end of the continuum and 10 being at the other end.3. Rave. students place each sport nearestthe group that enjoys it the most.can discuss their choices.StudentsYOUNG4. Rave students investigate the parks andrecreation program and facilities in yourcity. Row do the activities offered fitalong these continunms? Are there changesthat could be suggested that would improvethe program and/or facilities?ALLAGES383

The time of day, the temperature and the weather all have an effect on the sports oractivities you can participate in. Try to fill in one athletic activity that can bedone in each of the spaces of the clock, the thermometer and the climate zones. Somewill require some thinking.(f&~tt(;'iY'"';'''.f:~:'::":::='.12 f\lOOkl""""i~j~~t:~l;;.:::':': '.:il-__......::40• C(100" F)11- 35"'1- 30 ",1-- 2.5'11- 2.0"'.~~\~t0~{*t\;,,,,G PM.:.',.;..:./0J~',d!iiitJ,~r~l~1- 15"'1- 10 "5°SUklt-ly1--__ - 5 °11-- -10", ..• •. o' ",'" • '•• ~. • •~~::"":"•• " • c> - 0;> _" •"':'-----0 • ~ .. 1.'---...--,~ ..,1-- -15";1-__--'-20·(-10"\'")"'~OINV384 IDEA FROM: The Nature of RecreationPermission to use granted by M. l. T. Press

R '1Check the time distance ot each of these places from your home. Use the closest oneto your home. Indicate whether the time is by car (c), by bicycle (b) or walking (w).10 min 20 min 30 min 1 hr 2 hrs 3 hrsmore than3 hrsbasketball courtswimming poolparkballfieldriverski slopeskating rinkfishing lakecampgroundgolf coursebicycle pathbowling alleytracktennis courtsothers:The concentric circles on the next page have been drawn so that each .5 centimetresrepresents 10 minutes. Locate each of the places above by putting a dot in the correctdirection and with the correct travel time from your home.IDEA FROM:The Nature of RecreationPermission to use granted by M. I. T. Press385

R 7(COIJTIIIlUW)(\___ 3HREZ I-lR11-lRExample:1Point X is CZ hours in a south direction from your home.(386IDEA FROM:The Nature of RecreationPermission to use granted by M. l. T. Press

Which ball bounces the highest?Materials Needed: Any of the following: soccer ball, table tennis (ping pong) ball,basketball, tennis ball, volley ball, softball,golf ball.metre stickmasking tapeI. Select one of the balls for thefirst experiment. Attach the metrestick to a table or desk, so thestick stands vertically.10ClcmI,,,,o CJYT'­II. Drop the ball from the 100 emmark. Have your partner measurehow high it bounces. Read the markto the nearest em at the bottom ofthe ball. Record on this table.Drop the ball from the other heightslisted and record the bounces. Repeatseveral times to get accuratereadings.1009080BALL50I~ 4030§ 20w:r 10o o to 2.0 30 40 50 GO 'TO 80 SO 100I4EIGW Bl\LLDROPPEDI11,11-, 1, ..~~1, I4EIGI4T,, AT FIRST11 BOUNCE, 11 ,1 I, I METRE, 1STIClZ" 1V. Are therethe pattern?error?I4EIGI4T I1EIGI-FBALL BALLDROPPED BOUl\lCED10080GO5020III. Plot the points of your table on thisgraph.IV. The points probably do not fit in aperfectly straight line, but they do followa pattern. Draw a BEST-FIT line.Start at the zero-zero point and try tokeep as many points above the line asbelow it.any points that don't fitWhat could have caused anUse the table to predict the bounceof a ball dropped from 90 em, from 200 em.VI.Try the same experiment with other balls, recording, then graphing the bounces.VII. Which ball has the highest bounce on most of the heights? Do any of the balls producesimilar bounces?387

Find a soccer ball. The Physical Education teacher may have one you can borrow.A soccer ball is a three-dimensional mathematicalmodel called a polyhedron.VERTICES(a) The ball is made using two different polygonsfor faces.What are the 2 shapes?How many of each shape are on a soccer ball?EDGES """"~-How many total faces are on a soccer ball?b) A vertex is a point where 3 of these faces meet. Look at one vertex of the soccerball.How many 6-sided faces touch that vertex?How many 5-sided faces? (Look at other vertices and count the polygons that touch each one. Describe yourfindings.Count the vertices on the soccer ball. How many are there?How can you figure a way to get the total without counting all of them?c) Edges are formed when any two of the polygons meet. Count the edges to find thetotal number on the soccer ball.How can you make sure you don't countan edge twice?d) Imagine an insect crawling along the edges of the soccer ball. Could it crawl alongeach edge and return to its starting point, without crawling on any edge more thanonce?388(

5ItR5E5Mateh up the following balls with a size and mass given below.a mass in the square of the ball that you assoeiate with them.Write a diameter andBiLLIARD BALL SHOT PUTBASKETBALLGOLFBALL$"ifitIIBOWLlIJGBALLTABLE TENt-l\S BALL(PltJG POIJG)oBASEBALLVOLLEVBALLI-\AIJD•BALLSOCCE.RBALLTEIJI\l\S BALL SOFTBALLDiameters:Mass:g = gramskg = kilogramsa) 3.8 em g) 9.7 em 1) 2.4 g 7) 177 gb) 4.3 em h) 11-13 em 2) 28 g 8) 200 ge) 4.8 em i) 21 em 3) 46 g 9) 450 gd) 5.7 em j) 21.8 em 4) 57 g 10) 595 ge) 6.5 em k) 22 em 5) 65 g 11) 4.5 kg to 7 kgf) 7.3 em 1) 24.3 em 6) 140 g 12) 4.5 kg or 5.4 kg389

( COI\lTllJUE.D)5E5The following questions, based on the information from the previous page, could beused with your class.to promote discussion.Some of the questions are entirely hypothetical and are presentedUse them according to the interests of your class.(1) Which of the measurements, diameter or mass, shows a greater range?2) Does the ball with the largest mass also have the largest diameter?3) Does the ball with the smallest mass also have the smallest diameter?4) Name two factors that influence the mass of a ball?5) Classify the balls into three groups according to the mass.Light Medium Heavy6) Imagine that all the large balls had proportionally the same mass as a bowling ball.What would be some problems if such a ball were used to play soccer, volleyball orbasketball?7) What would happen if you played golf with a table tennis ball?(8) Rotation is a game of billiards where a person's score is determined by finding thesum of the numbers of the balls that person hits into the pockets of the table.What is the fewest number of balls that can be hit into the pockets to assure aperson of winning in a two-person game?9) Examine the polygons that make up the surface of the soccer ball. Name the twotypes.How many of each type are there?10) How many pieces of material are sewn together to make a baseball or softball?Are the pieces congruent? Make a sketch of one of the pieces. How manylines of symmetry does the piece have?How many seams are needed to sew thepieces together?(390

The volume formula for a sphere (ball) is:OR 1.3'5 '3.1"+ .,..~\, xrOR 1+.18' Y • r x YGiven the diameters of some balls, find their radius, then their volume.TABLE TI;.NNIS BALLBASEBALLSOCC£R0 ®BALL@"""iidiameter = 3.8 em diameter = 7.3 em diameter = 22 emradius = '! radius = i.ii ,;: -radius = " ('ii,volume = i)!). " volume = "i" " volume = !',i,; i "i",'•0I-lA\\lDBALL GOLF BALL,"Lccv-«DBALL ;,. "'''''U"'~BALL,'.;;... ::~:. :':diameter = 4.8 em diameter = 4.3 em diameter = 21 em diameter = 21.8 emradius = radius =, ' , radius = radius = " '.: ",volume = volume = ., "" volume = c ," volume = '''i', ) ,,"5HOT PUT BILLIARD BALL ,£:\\lNIS BALL(J G) Q)"""n-~~',' .:.:? BALL :,.",. ""':'~:i ·ti:~~diameter = 12 em diameter = 5.7 em diameter = 6.4 em diameter = 24.4 emradius = radius = .' r,' radius = : ) ,radius =') ) ),volume = ','!) , ...., volume ,..= '::'. ivolume = i ,ii volume" = ,cc ) .,.,;,'Finding the volume of a football is much more difficult. It's shape is called a prolatespheroid. A football has a long diameter of 53 em and a short diameter of 28 em. The~ireum£erenee around the fattest part of the football is about 71 em. Do you have anysuggestions for finding the volume of a football?391

3925.This activity consists of ten events foryou to perform.The best performance in eachevent will be scored 100 points with the remainingscores being the appropriate percentof the best performance.The ten events of this decathlon are:1. Cotton ball throwThrow a ball of cotton as far as you can. Measurethe distance from the starting line to the pointwhere the cotton ball first touches the floor.2. Basketball putIn the gym or outside throw a basketball as far asyou can. Use a motion like a shot putter. Measurethe distance from the starting line to the point thebasketball first touches the floor.3. Standing long jumpJump as far as you can from a standing start. Mea~sure the distance from the starting line to the backof your heels where you jump.4. Tiddley~wink snapSnap a tiddley-wink as far as possible. Measure thedistance from the starting line to the point wherethe tiddley~wink first touches the floor.Twenty-metre, one-legged dashIn the gym or outside record the time it takesleg up off the floor.to hop 20 metres while holding one6. Plastic straw throwThrow a plastic straw as far as possible. Measure the distance from the startingline to the point where the straw first touches the floor.7. Fifty-metre backwards runOutside record the time it takes to run 50 metres backwards.8. Airplane flyIn the gym or outside throw a paper airplane as far as you can. Measure the distancefrom the starting line to the point where the plane first touches the floor.9. Rubber band snapSnap a rubber band as far as you can. Measure the distance from the starting lineto the point where the rubber band first touches the floor.10. Paper toss scoreFrom a distance of 4 metres record how many wads of paper you can throw, 1 at atime, into a wastepaper basket in 30 seconds.Pick a partner.For each of the ten events, as one person is performing, the partnerwill be either timing or measuring distances. (Each person will receive three separate trials in each event.trials on your individual record sheet.Label and record the(

PERSONAL RECORDS OF CLASSROOM DECATHLONNAME:PARTNER:Trial 1Trial 2 Trial 3EVENT:EVENT:EVENT:EVENT:EVENT:EVENT:EVENT:EVENT:EVENT:EVENT:Circle your best performance for each event.393

(PERSONAL RECORD OF CLASSROOM DECATHLONFinal Summary CardNAME:PARTNER:EVENT: BEST: SCORE:EVENT: BEST: SCORE:EVENT: BEST: SCORE:EVENT: BEST: SCORE:EVENT: :BEST: SCORE:(EVENT: BEST: SCORE:EVENT: BEST: SCORE:EVENT: BEST: SCORE:EVENT: BEST: SCORE:EVENT: BEST: SCORE:The best performance in each event is assigned a value of 100 points.scores are figures on what percent they are of the best performance.The remainingFor example, if thebest distance in the cotton ball throw is 8.0 metres, it would be worth 100 points.score of a 6.5 metre throw would be as follows:

Get your measurements for the following.foot lengthin centimetresheight incentimetresrb:::::::::"Jl--:::::==ojarm stretchin centimetresarm reachstandingflat-footedin centimetresstanding longjump incentimetresstanding highjump incentimetresol. What is the ratio of youra) arm stretch to heightb) foot length to heightc) arm reach to heightd) standing long jump to heighte) high jump to height~f) standing long jump to high jump2. Write each of the above ratios as a percent. Put the answers in the blanks above.3, Use the information from your entireclass and make a graph for each of thepercents. Do any of the graphs show apattern?4. Can you predict the approximate highjump of a person if you know the personftheight? How about foot length?120%110%100%90%80%70%GO%50%40%30%2.oro10%r(COMPARISOtJ OF ARM SiRETO-l 10 \.lEIGHT.~SAM JUAII\ JOE BET~ CAROL S\.lER\(()395

Most sporting activities involve an extensive use of muscles. Usually thoseathletes who have conditioned their muscles through practice perform better and haveless muscle fatigue. In this activity students will measure muscle fatigue.(Three students are needed per group; one as the "muscle" person, one as the timer,and one as the counter.The "muscle" person places the right forearm flat on a table so the back of thefingertips are flat on the tabletop. He/she closes and opens the right hand as fastas possible until the timer says stop, being sure the fingertips touch the palm whenclosed and the fingertips touch the table when open.The timer times each trial for 30 seconds and records the tally on the chart.Between the 3rd and 4th trials the "muscle" person is given a 30-second rest period.The counter counts the number of times the fingertips touch the table, relays thecount to the timer, and begins the count over at 1 for each trial.After the activity has been completed for the right· hand, repeat the steps forthe left hand.8030 SECO}JD RIGHT U::f'TPERIODS HAl\}O \.lAND1 72- (;82- fOO GO3 4'7 45REST REST REST4 50 52,0GO504030RIGHTLEF'T~~ /"-~ ,.......... "-';."'"/~ ~~(5 J..l.2.. 45G 3'7 352.010RESTSample questions could include:1) What is the amount of decrease froma) 1st period to 2nd periodb) 1st period to 3rd periodc) 1st period to 6th periodo 3 4- 5 G(3962) Find the percent of decrease for each of the parts of (1).IDEA FROM: Mathematics and Living Things

w= ~-My HEARTTHRO S~12. Yo~Materials Needed:2 studentsStopwatch or clock with a second handActivity: 1. On your paper draw a chart like the one below.Ino..ctive ActiVe. T2€.co'!e'IVNo.me Pu.lse. "Pu.lse. Pulse8

(Numerous ads to eat wisely and exercise regularly encourage students to thinkabout their physical condition which, in turn, affects the pulse and recovery timefollowing exercise. In general, conditioned persons have a slower resting pulse anda slower pulse during exercise. Their pulses will recover to the resting ratequicker following strenuous exercise than the pulses of persons who are in poor condition.Because of heredity, some persons inherit efficient hearts with slowerrates, while others are born with relatively inefficient hearts. However, bothtypes can be improved.Since the physical condition of an individual affects his heartbeat, pulsetests can be used to measure physical fitness. Four pulse tests are describedbelow, and tables to interpret the results are provided. Better results could beobtained from the first two tests if they are done at home with parental help.1. Pulse Lying:The pulse lying is the slowest, resting pulse of a person. The student canfind this rate by taking her pulse for 30 seconds befo~e she gets out of bed inthe morning. If done in class, have the student lie down and attempt to completelyrelax for ten minutes. In the lying position count her heartbeats for 30 seconds.The student should continue to rest in the lying position for 2 more minutes andrepeat the count. If it is the same double the count to get the pulse lying, andrecord the number. If less the student should rest longer and repeat the count.II. Pulse Standing: (To obtain the slowest, resting, standing pulse have the student rise slowlyafte.r finishing the pulse lying test and remain standing for two minutes. Countthe heartbeats for 30 seconds and double the number to get the pulse standing.Have the student subtract the pulse lying from the pulse standing. Thisnumber is the pulse difference. By checking Table A the student can find her physicalfitness rating.398TABLE AAPhYS;CQ! /:Jf?);4;.~~p~'YFi tness vit'li ::::..tb..(~.l ~7 5'8 60 G3 66 69 71 73 7S 77 78 79 80 82 84 86 105Pu.lse Sta.ndir'\Q 410 63 67 G8 70 74- 77 80 83 85 87 90 9\ 92 94 96 98 101 IZ3~9 10 \2 12- 12 13 13 13 /4 14 14- 15 (8Pulse Ditterehce 6 10 /0 II 1\ IIIDEA FROM:Physical Fitness Workbook, by Thomas Cureton,Copyright © 1944 by Stipes Publishing CompanyPermission to use granted by Stipes Publishing Company(

III.(CONTll\IU..D)Simplified Pulse Ratio Test:(a) While sitting, have the student count and record her heartbeats forone minute.(b) Have the student face a chair (approximately 45 cm high) and step upwith the left foot, up with the right, down with the left, and downwith the right. The student should do 30 of these steps in one minute.In order to set the cadence the teacher or another student can callout "up, up, down, down" at the required speed or playa taped recordingof the cadence.(c) Immediately after completing the 30 steps, the student should sit,count and record her heartbeats for two minutes.(d) Have the student write her pulse ratio. Pulse Ratio = Heartbeats for2 minutes following the exercise : Heartbeats for 1 minute beforeexercise. Simplify theratio by dividing the firstnumber by the second, correctto one decimal place. TABLE BCheck Table B to find thephysical fitness rating.PulsePhysica.lRa..tio Fi+ne!i,s, R.a:ti 119IV. Three Minute Step Test:This test is administered likethe previous test, except thestudent steps for three minutes,and the cadence is 24 steps perminute. In addition wait oneminute after the student completesthe exercise and countthe heartbeats for only 30seconds. The efficiency scoreis the ratio ofnumber of seconds stepping x 100pulse for 30 seconds x 5.6Divide and check Table C to findthe physical fitness rating.*This table is accurate for juniorhigh girls. The efficiency scoresmay need to be raised for juniorhigh boys. At the grade schoollevel there is not much differencebetween boys and girls.Below 2.'3 Above AverQfije2.3 - 2.'7 Avero.'ijeAbove 2.'7TABLE C*Below Aver

vvhors YOUQ TYPE1. Weigh yourself and measure your height. ___ pounds inches(2. Change your weight to kilograms.1 pound ~ .45 kilograms.3. Change your height to centimetres.(1 inch"" 2.5 centimetres'J4. Use the chart to determine your body.45 kilogl'Qvn'31 pou.V\d~-/'---1type.? ----Ma.o>o> in kilc'jramsHei'jht in centimetres GROWTH CHAR.I FOR. GIRLS10 Yrs. II Yys. \2. Yrs. \3 Yrs 14 Yrs. 15 VI's.Ta.l I 143-155 153-IG3 157-168 IG2-170 1102- 173 IG4 -173Avera.ge 134-\42 140-152 147-156 152-IG\ 154-IG\ 156-163Shol't 125-133 130-139 \35-146 140-151 146-153 147-155{ H,uv 40-52 45-59 49-63 55-68 57-71 60-72-Avera.qe 29-39 33-44-.36-48•...•4\ -54 45-56•.47-59Liqh+ 2.3-2.8 25- 32- 2.8-35 31-40 36-44 39-46. . - .GROWTH CHART FOR. BOYSTa.1I 149- 155Aveya.qe 134-148{Shov-t 1:

Estimate the number of hours (to the nearest half hour) per day that you do each of theactivities below. Find the number of Calories you should eat to provide the energy todo these activities.Calories used per kg Your body CaloriesActivity of body mass per hour Time in hours mass usedBicycling (fast) 7.5Bicycling (slow) 2.4Dishwashing 1.1Dressing/Undressing .7Eating .4Playing Ping Pong 4.4Running 7.3Sitting Quietly .4Sleeping .4Standing .4Studying .4Swimming 7.9Tennis 6.6Typing 1.1Volleyball 5.5Walking 2.0Work, heavy 5.7Work, light 2.2How do your total Calorie needs compareto the chart to the right which shows theCalories needed for boys and girls ofaverage height and mass for one day.TotalCALORIES NEEDED9·12 YRS. 12.-15 YRS.BOYS 2Lf.OO 3000GIR.L'S 2200 '2.

AnRRI; L,JOUPW~~I[~LLWr=IT'?interesting and fun activity (perhaps in cooperation with thephysical education teacher) would be to test the physical fitness of yourstudents.The President's Council on Physical Fitness has suggested thefollowing seven tests: pullups, situps, standing jump, shuttle run, fiftyyarddash, softball throw for distance and six hundred-yard run/walk.(Standards of performance for boys and girls, ages 10-17, are shownbelow. Notice that the charts also include standards for qualifying forthe Presidential Physical Fitness Award. Official applica-tion forms andcomplete information can be obtained by writing to Presidential PhysicalFitness Awards, 1201 Sixteenth Street N.W., Washington, D.C. 20036.The data collected during testing can be used for motivating severalconcepts of descriptive statistics such as mean, median, mode, range,etc. The data can be used to make several kinds of graphs, e.g., a bargraph comparing performances of students in a class or a line graph torecord progress of an individual student. Students will probably be very (interested in comparing results from earlier tests to see the progressthat has been ll\ade.PULLUPSA boy grasps the bar withpalms forward and hangs fromthe bar so his feet don'ttouch the floor. A successfulpullup is completed whenthe boy pulls his body upuntil his chin is over thebar and then lowers his bodyuntil his elbows are fullyextended.FLEXED ARMHANGTwo spotters help a girl toraise her body so her chin isabove the bar, palms forward,elbows flexed and feet off thefloor. A timer records the time,in seconds, that the girl canhold this position.402RatingBOYS[Number of pullt\ps]10 11 12_1_3_1~15 16 17---- ------Excellent. ...... , .... , . 8 8 9 10 12 13 14 16Presidential Award.•.... 6 6 6 8 10 10 12 12Good..... ............. ' 5 5 5 7 8 10 II 12Satisfactory..... 3 3 3 4 6 7 9 9Poor .. I I I 2 4 5 6 7RatingGIRLS[In seconds]10 11 12 13 14 15 16 17-- --Excellent .. 31 35 30 30 30 33 37 31Presidential Award.. 21 20 19 18 19 18 19 19Good .. 18 17 15 15 16 16 16 16Satisfactory.. 10 10 8 9 9 10 9 10Poor .. 6 5 5 5 5 G 5 GIDEA FROM:Youth Physical FitnessAgeAge(

BOYSSITUPSA pupil lies on his backand locks his fingers behindhis head while d partnerholds his ankles to keep hisheels on the floor. A situpis completed when the pupil(1) sits up, (2) touchesright elbow to left knee,(3) lies back down, (4) sitsup, (5) touches left elbowto right knee and (6) liesback down.Rating[Number of situps]10 I11 12 13 14 15 16 17-------------Excellent, ..... ..... 100 100 100 100 100 100 100 100Presidential Award.. 100 100 100 100 100 100 100 100Good, .. . .. ... ... . . . ... 76 89 100 100 100 100 100 100Satisfactory..... .. .. ... 50 50 59 75 99 99 99 35Poor. 34 35 42 50 60 61 63 57--GIRLSExcellent. ...... 50 50 50 50 50 50 50 50Presidential Award. 50 50 50 50 50 50 50 50Good.... 50 50 50 50 49 42 41 45Satisfactory........ 39 37 39 33 34 30 30 30Poor. ., ...... 26 26 26 27 25 24 24· 23AgeSTANDINGLONG JUMPA pupil stands comfort~ably with knees flexed andthen jumps forward as faras possible. The distanceis measured from the startingline to the point wherethe heels first touch thefloor.RatingBOYSAge10 II 12 13 1415 16 17---------------------ft. in. ft. in. ft. in. ft. in. ft. in. ft. in. ft. in. ft. in.Excellent........ 6 I 6 3 6 6 7 2 7 9 8 a 8 5 8 6PresidentialAward...•... 5 8 5 10 6 2 6 9 7 3 7 6 7 11 8 1Good..•........ 5 7 5 9 6 I 6 7 7 a 7 6 7 9 8 0Satisfactory...... 5 2 5 4 5 8 6 a 6 7 7 a 7 4 7 6Poor........ , .. 4 10 5 0 5 4 5 7 6 I 6 6 6 II 7 0GIRLSExcellent..... , .. 5 8 6 2 6 3 6 3 6 4 6 6 6 7 6 8PresidentialAward ....... 5 4 5 8 5 9 5 10 6 0 6 1 6 2 6 2Good..... , .... 5 2 5 6 5 8 5 8 5 10 6 a 6 0 6 0Satisfactory.... 4 10 5 0 5 2 5 3 5 5 5 6 5 6 .> 7Poor .. , ... , .... 4 5 4 8 4 9 4 10 5 0 5 I 5 2 5 2403

SHUTTLERUNTwo blocks of wood, 2"x2"x4," are placed 30 feet froma starting line. A pupil runsto the blocks, picks up oneblock, runs back to the startingline and places the blockbehind it. This action isrepeated to place the secondblock behind the startingline. The time needed toaccomplish the task is recordedto the nearest 10th of asecond.qR~ I;JOUPW~C:I[~LL~~IT?(PAGE. 3)RatingExcellent. .....Presidential Award.Good , .Satisfactory , .. , , ...Poor .BOYS[In seconds to nearest 1Ot11J10 I 1110.010.410. 5I!. 0I!. 512 I 13 1 14Excellent. 10.0 10. 0 10.0 I 10. 0 10.0 10. 0 10.0 10.0Presidential Award. 10.8 10.6 10.5 10.5 10.4 10. 5 10. 4 10.4Good......... I!. 0 10. 9 10. 8 10. 6 10. 5 10. 7 10.6 10.5Satisfactory ...•. ...... . 11. 5 11. 4 11. 3 11. 1 11. 0 11. 1 11. 0 11. 0Poor........... . . . . ... 12.0 12.0 11. 9 I I!. 8 I:~: ~ 1~: ~ I ::~ I :: ~ :: ~ :: ~ ~: ;10. 4 10. 2 I' 10. 0 i 9. 8 9. 5 9. 3 9. 210.9 10.7 10.4 I' 10.0 9.8 9.7 9.611. 3 I!. 1 110. 9 I 10. 5 10. 1 10. 0 10. 0GIRLS1Age151617J 1. 5 11. 6 11. 5 11. 5((50-YARD DASHThe time needed for apupil to run 50 yards isrecorded to the nearest10th of a second.RatingBOYS[In seconds to nearest 10th]10 11 12IAge13 14 15 16 17- ------1-'---------Excellent.•.';... .,.,.,." 7.0 7.0 6. 8 I 6. 5 6. 3 6.1 6.0 6.0Presidential Award...... 7.4 7.4 7.0 ! 6.9 6.6 6.4 6.2 6. 1Good................. 7.5 7.5 7.2 7.0 6.7 6.5 6. 3 6.2Satisfactory...... , .. , .. 8. 0 7.8 7.6 7.3 7.0 6.7 6.5 6.5Poor..... , ....... , .... 8.5 8.1 8. 0 7.6 7.2 7.0 6.8 6.7iIGIRLSIExcellent... , .. .... 7.0 7.0 7.0 7.0 7.0 7.1 7.0 7.1Presidential Award. 7.5 7.6 7.5 7.5 7.4 7.5 7.5 7.5Good....... ," , 7. 7 7. 7 7.6 7.6 7.5 7. G 7,5 7.6Satisfactory..... 8.2 8. J 8.0 8. 0 7. 9 8. 0 8.0 8.0Poor. , ... , " , . ", .. , .. 8.8 8.5 8.4 8.4 8.3 8. 3 8.5 8,5(404

BOYSSOFTBALL THR0\4A softball is thrown asfar as possible with an overhandmotion. The best ofthree throws is recorded bymeasuring the distance fromthe starting line to thepoint where the softballfirst touches the ground.Rating[In fcct]Age10 ' 11 12 13 14 15 16 17----Excellent. , .. 138 151 165 195 208 221 238 249Presidential Award.. ... 122 136 150 175 187 204 213 226Good.... , " . ... 118 129 145 168 181 198 207 218Satisfactory. ... . .. ... 102 115 129 147 lG5 180 189 198Poor. 91 105 115 131 146 lG5 172 180GIRLS.Excellent. ... 84 95 103 III 114 120 123 120Presidential Award.... 71 81 90 94 100 105 104 102Good.... 69 77 85 90 95 100 98 98Satisfactory..... " ' " 54 64 70 75 80 84 81 82Poor. .... ... " ' " . ... 46 55 59 65 70 73 71 71GOO-YARDRUN/WALKThe time needed for apupil to run 600 yards isrecorded in minutes andseconds. Walking is permitted,but the objectiveis to cover the 600 yardsin the shortest possibletime.RatingBOYS[Iu minutes and seconds}Age10 11 12 13 14 15 16 17--------------Excellent, . , ......... 1:58 1:59 1:52 1:46 1:37 1:34 1:32 1:31Presidential Award... 2:12 2:8 2:2 1,53 1:46 1:40 1:37 1:36Good............... 2:15 2:11 2,5 1:55 1:48 1:42 1:39 1:38Satisfactory.......... "2,26 2:21 2,15 2:5 1:57 1:49 1,47 1:45Poor .... , ..... , . ,. 2:40 2:33 2,26 2,15 2:5 1:58 1:56 1:54GIRLSExcellent. , ........... 2:5 2:13 2:14 2:12 2,9 2:9 2:10 2:11Presidential Award .... 2:20 2:24 2:24 2:25 2:22 2:23 2,23 2:27Good................. 2:26 2,28 2:27 2:29 2,25 2,26 2:26 2,31Satisfactory.......... , .. 2:41 2:43 2,42 2:44 2:41 2,40 2:42 2:46Poor ................... 2:55 2:59 2:58 3:0 2:55 2:52 2:56 3:0405

The table to the right shows the final medalcount for the 1976 Olympic Games.1) What was used to place the teams in theorder that they appear?2) Find the total number ofa) gold medalsb) silver medalsc) bronze medals3) Why do you think the totals are different?Competitions are often scored by awardingpoints for places.Final medal countCountrYSoviet Union.East Germany .United States.West Germany ;Jopan.. 'Poland,BulgariaCuba..RomaniaHungarvFInland.Sweden.Britain.Italy.' .France.YugoslavIa . .Czechoslovakia .New Zealand.Sooth Korea .SwitzerlandJamaicaNorth Korea.Norwav.DenmarkMexico.TrinidadCanada.BelgIumHolland.PortugalSpainAustn:illoIron. .MongolloVenezuela.Brazl! .Austria.Bermuda .Pakistan .Puerto RIcoT~alfandGold Sliver Bronze Total47 43 35 12540 25 25 9034 35 25 9410 12 17 39961025876811925246 .4 :1 13491427.4 5 12 214 :2 0 64 1 0 5-l 5 5 13:2 7 -4 13:2 :2 5 9:2 3 :1 8:2 :2 4 8:2 1 1 .41 1 4 6J111~0.4:21111002:21 0 :2 31 0 J :21 0 0 15 6 113 3 62 3 52 0 22 0 21 4 51 1, :21 0 11 0 10 :2 :20 1 10 1 I0 1 I0 1 1o 0 I 1(4) If 3 points are awarded for a gold medal, 2 points for a silver and 1 pOlnt for abronze, would the total points awarded cause the order of the top five teams to (change?5) What happens to East Germany and the United States if the points are 5 for a gold,3 for a silver and 1 for a bronze?6) Find a reasonable total point system that places East Germany ahead of the UnitedStates.7) Find the order insystems are used.total points for Japan, Poland and Bulgaria if these scoringa) 3, 2, 1b) 5, 3, 1c) 7, 4, 18) Will any of the three scoring systems in (4) allow Italy to finish ahead of Britain?9) Project: Look up the population of each country in an almanac and find a people permedal ratio. Which country has the lowest people per medal ratio?Do you know any reasons why this might be so?(406

~JbW~ [MJ~ ~ rnJ~OO[QJ ~uillJ~uA model of a running track with the start-finish line is shown below. Use a piece of nonstretch stringto find out how much farther a runner in the outside lane runs than a runner in the inside lane. Use thisdistance to mark a new starting line for the outside lane so the distances from start to finish are the same.In a similar way find the new starting lines for the other lanes.~o-..j

(meanS 3 mi nutesI10 secondsWORLD RECORDS *1) 1 mile -.3:51.02 miles -'8:13.83 miles - 12:47.8Each of these three records isroughly proportional to runninga mile every minutes.2) Steve Williams of the U.S. ran the 100-metre dash in 10.1 seconds andthe 200-metre dash in 20.6 seconds. His speed was about metres per second.(3) Tommie SmithThe time foreach race?of the U.S. ran both the 200-yard dash and the 200-metre dash.both is 19.5 seconds. Does this mean he ran the same speed forIf one race has a faster speed which race is it?4) These are world records. Rank them from slowest to fastest based on the timetaken to go 1 kilometre (1000 metres).a) Canoeing (1000 m) 3:48.06b) Swimming (1500 m) Men 15:31.85c) Running (1500 m) Women 4:01.4d) Ice Skating (1500 m) Men 1:58.7e) Running (1000 m) 2:16f) Swimming (1500 m) Women 16:49.9g) Cycling (1000 m) 1:7.51h) Ice Skating (1500 m) Women 2:15.85) Is the world recoro of 9.9 seconds for 100 metres faster or slower than the worldrecord of 9.1 seconds for 100 yards?(1 yd. oz.9l4 m or 1 m ~ 1.1 yd.)6) The world land speed record for a jet-propelled automobile is about 622 miles perhour. At this rate how long would it take to drive to the moon (if there was aroad) 240,000 miles away?(408 *As of January 1, 1974 (The Official Associated Press Sports Almanac 7974)Permission to use granted by The Associated Press

WORLD TRACK RECORDS-FIELD EVENTS-HIGH JUMP 7'~" 8POLE VAULT18' .s+"4LONG JUMP 29'21."2TRIPLE JUMP 57' 2 1 "4SHOT PUT71' 7"DISCUSJAVELIN224'5"308'8"60'0"221'9"213'1. HOW MUCH HIGHER IS THE MEN'S POLE VAULT RECORD THAN THE MEN'SHIGH JUMP RECORD?2. HOW MUCH FARTHER IS THE MEN'S JAVELIN RECORD THAN THE MEN'SDISCUS RECORD?3, HOW MUCH FARTHER IS THE WOMEN'S DISCUS RECORD THAN THE~OMEN'S JAVELIN RECORD?4, HOW MUCH HIGHER IS THE WOMEN'S LONG JUMP RECORD THAN THEWOMEN'S HIGH JUMP RECORD?5. A PENTATHLON IS MADE UP OF FIVE EVENTS. SUPPOSE ONEHELD ALL FIVE OF THE ABOVE LISTED WORLD RECORDS. WHATWOULD BE THE TOTAL OF THIS "PENTATHLON"?\~OMAN6. HOW DO YOU THINK THE WEIGHTS OF THE MEN'S AND WOMEN'S DISCUSCOMPARE? WHICH ONE IS HEAVIER AND BY HOW MUCH?7. WHY AREN'T THERE RECORDS FOR THE WOMEN'S POLE VAULT AND THETRIPLE JUMP?• As of January 1, 1974 (The Official Associated Press Sports Almanac 7974)Permission to use granted by The Associated Press409

WORLD TRACK RECORDS-RUNNING EVENTS-220-YD.MILE6-MILE~~~s*MEN WOMEN100-YD.DASH 9.1 SEC. 10.0 SEC.DASH440(QUARTER MILE)20.0 (TURN) 22.644.5 52.2880 (HALF MILE) 1 MIN.LI4.6 SEC. 2:02.03:51.1 4:29.526:47.0(1.2.3.4.FOR THE MEN, HOW MUCH LONGER IS THE TIME:FOR THE LI40 THAN FOR THE 220?)/.i ).iFOR THE 220 THAN FOR THE 100?FOR THE 880 THAN FOR THE 440? M.FOR THE 6-MILE THAN FOR THE MILE?SUPPOSE YOU COULD RUN FOUR QUARTER MILES IN A ROW AT THE RECORDPACE FOR WOMEN.A) HOW LONG WOULD IT TAKE YOU TO RUN THE MI LE? iii;i ( u;. CCi:B) WHAT IS THE DIFFERENCE BETWEEN THIS TIME AND THE WORLD RECORDFOR THE MILE?;;i . ce;:SUPPOSE A MAN RAN SIX MILES IN THE RECORD TIME AT AN EVEN PACE.A) HOW LONG WOULD HE TAKE TO RUN ONE MI LE? c ceB) WHAT IS THE DIFFERENCE BETWEEN THIS TIME AND THE WORLD RECORDFOR THE MILE? enTHE BOSTON MARATHON IS 26 MILES, 385 YARDS. ESTIMATE THE RECORDTIME FOR THE RACE. CHECK YOUR ESTIMATE IN AN ALMANAC.410FROM: The Official Associated Press Sports Almanac 1974Permission to use granted by The Associated Press(

HOW THE WORLD RECORD PROGRESSED100 MetresLong JumpShot PutHigh JumpJimThorpeUSA191211.222-:t1:-442-3l 26-118PaavoYrjo1aFinland192711. 722-1346-9 476-- 8BobMathiasUSA195210.922-1of150-2­ 276-2­ 8RaferJohnsonUSA196010.624-~ 452-05-1~ 8KurtBendlinW. Germany196710.624-~ 447-76- !. 2BillToomeyUSA196910.3125-5­ 2147-2­ 46-4Niko1ayAvi10vRussia197211.00125-2­ 2147-1­ 216-11- 2400 Metres52.252.850.248.647.947.148.5High Hurdles15.616.814.714.514.814.314.31DiscusPole VaultJavelin121-410-8140-11133-~10-6188-4153-10113-1 2194-3170-~113-- 43233-2­ 4151-111 413-4 2245-71152-6­ 4114-- 4215-8154-11 214-111 4202-3f1,500 Metres4:40.14:54.24:50.85:09.94: 19.44:39.44:22.8Total67566,7687,7318,0638,3198,4178,451The above represents only 11. sampling of world record decathlons for purposes of illustratingadvances in the event and should not be taken as a complete list.a) How many events take place at a decathlon meet?b) They are all part of what general sport?c) A player competes against the existing records for each event and a record-tying orrecord-beating performance is worth 1,000 points. How many points could a decathletepossibly be awarded? __._.. _--'- _d) In general, have the times for ru~ning events increased or decreased? Have the distancesfor field events increased or decreased?e) Who ran the fastest 1,500 metres? the fastest 100 metres?f) The Olympic records for the 100 metres was 10.8 in 1912, 10.4 in 1952, 10.2 in 1960,and 10 14 in 1972 Which decathlete was closest to the Olympic record at that time?g) Find out who the current world record holder is. What are his times and distances?h) How many events would occur at a pentathlon? What are they?411

(Rules for bowling with dice:1) Roll both dice.a) Lf the sum is 10 or more, record a strike.b) If the sum is less than 10, record the sum as the number of pins knocked down onthe first ball.2) Roll one die.a) If the number shown plus the number in (b) above is 10 or more, record a spare.b) If the number shown plus the number in (b) above is less than 10, record thenumber as the pins knocked down on the second ball.3) If several people are playing, take turns rolling the dice.(412

S[DRt~ttPINbIN BDWLINbC15=JIn bowling a person throws the ball two times each frame.Three things can happen.1) Open - some of the pins knocked down with the first ball and some (but notall) knocked down with the second ball. See frames 1, 2, 4 and 9.2) Spare - some of the pins knocked down with the first ball and all the restknocked down with the second ball.See frames 3, 5 and 8 in the sample game.3) Strike - all the pins knocked down with the first ball, the second throw isskipped. See frames 6, 7 and 10 in the sample game below.Points are earned for each frame as described below.are computed,frame.After the points for each framethe points for that frame are added to the total score in the previousThe new total score is then recorded.1) Open your score for the frame is the total number of pins knocked down.Example:.of 7 for5 pins down on first ball and 2 down on second ball--scorethe frame.2) Spare your score for the frame is 10 plus a bonus of what you knock downon the first ball of the next frame.Example:Spare in the frame 1 and 7 pins down on first ball inframe 2--score of 10 + 7 = 17 for frame 1.3) Strike - your score for the frame is 10 plus a bonus of what you knock downon the next two balls you roll.Example:An entire game is shown below.Strike in frame 1, strike in frame 2, 8 pins on first ballin frame 3--score of 10 + 10 + 8 = 28 for frame 1.frame are not shown on a regular score sheet.The rows with the names and the score for the individual~E 1 2 3 4 5 6 7 8 9 10 TotalNAMElID nEE !IDZ I.m

1) A bowling ball can have a circumference of not more than 68.6 cm. What would be thediameter?(2) A bowling pin is 38.1 cm high and has a base diameter of 5.7 cm. What is thecircumference of the base?3) The diameter of the "fattest" part of a bowling pin is twice the base diameter.What is the circumference of the "fattest" part?4) When positioned correctly, two adjacent bowling pins are 30.5 cm apart from centercenter. Show that if a bowling ball is rolled exactly between the two adjacentpins, the ball will (a) knock down both pinsor (b) roll between both pins.5) From the foul line to the first pin is a distance of 18.3 metres. How many timeswill a bowling ball tum over in this distance? (Assume the ball starts rollingat the foul line and does not skid.)6) A bowling alley from foul line to the end of the alley is 19.2 metres long. Thewidth of the alley is 106.8 cm. If a bowling establishment has 36 alleys, what isthe total surface area of all the alleys?(7)The bowling pins are numberedlike the diagram to the right.By moving just 3 pins, makethe pins tlpoint ltdirection.in the opposite.~GG0880808 0)8) Draw an arrow to show where the bowling ball should go to knock down all the bowlingpins in these diagrams.(a) (b) (c)o88(d)0)o (0(414

81:To find a baseball player's slugging percentage divide the "times at bat" intothe total base the player has.A single is worth one base, a double twothree bases and a home run is worth four bases.Here is an example of how to figure totalbases.In 1921 Babe Ruth hit 85 singles,-\ 1/-44,.-..,-dO_U--;bl""e,.-s'_/",1\16 triples and 59 home runs.85 singles is 85 bases44 doubles is 88 bases (2 x 44)16 triples is 48 bases (3 x 16)59 home runs is 236 bases (4 x 59)total 457 basesRUTH;" TOTALor "157 TOTALBASES STILLSTAtJD':> A'::. ARECORD FORMOST TOTAL BAS;';;l;;;S;;\'--\+---jr:--­ItJ A SltJGL" Sl;:ASOtJItJ TI-ll;: MAJORLl;:i\(;U£'3.Calculate the slugging percents for the following baseball players.Home Total SluggingPlayer At Bats Singles Doubles Triples Runs Bases PercentsMcGraw 352 70 23 11 26Franklin 292 37 16 5 12Hoffer 470 58 21 13 22Brannan 468 63 31 18 27McKenney 204 28 19 6 6Babe Ruth has the highest lifetime slugging percentage with .690 in 21 years withthe major league.417

YOU WILL NEED AN ALMANAC.BASEBALL AUTO AAc.lN6RECQRQ5USING THE INDEX} LOOK UP THE PAGE FOR BASEBALL--MAJOR LEAGUE LIFETIMERECORDS} LEADING BATTERS (OVER 2}000 HITS).Name of batterCobb , TyHornsby,RogersDelehanty, EdWilliams, TedRuth , BabeGehrig , LouSisler, GeorgeMusial , StanDiMaggio, Joebatting averageUsing the index, look up the page for AUTO RACINGaverage rounded offto hundredths .•••••(Indy.5DO)((YearWinnerCarwinning speedin miles/hourm.p.h. roundedoff to tenths.19111920193419391947195319591961196519691972(418SOURCE: Project R-3Permission to use granted by E. L. Hodges

Each week during the I6otball)~~ason collect the scores involving thelocal high school team. You may wish to use professional or college scores.Put them together at the enddents do the following.of the season (see sample below) and have stu-1. Place the winning scores in order--largest to smallest.2. Make a HISTOGRAM of the winning scores. Use graph paper.3. Find the highest winning score.Find the lowest winning score.Find howfar apart the highest and lowest scores are.____(RANGE)4.Determine the most frequent winning score.(MODE)5. Determine the middle winning score. (MEDIAN)6. Determine the average winning score. (MEAN)7. Arrange the LOSING scores in order--largest to smallest.8. Answer 3, 4, 5 and 6 for the losing scores.9. Were there any games where the winning and losing scores were the sameas the median winning and losing scores?District 5AAA Football ScoresWeek 1Thurston 28, Sheldon 8Marshfield 21~, Cottage Grove 12North Bend 34, Churchill 18South Eugene 26, Willamette 8Springfield 12, North Eugene 7Week 4Churchill 25, Springfield 0Marshfield 13, Sheldon 12South Eugene 34, Cottage Grove 26Thurston 16, North Eugene 14North Bend 40, Willamette 26Week 2North Bend 22, Sheldon 20Churchill 21, South Eugene 20Willamette 29, North Eugene 8Cottage Grove 40, Springfield 18Marshfield 59, Thurston 28Week 5North Bend 42, North Eugene 20South Eugene 36, Marshfield ,15Cottage Grove 7, Thurston 0Sheldon 12, Springfield 7Churchill 35, Willamette 21Week 3North Eugene 14, Cottage Grove 7Sheldon 22, South Eugene 21Thruston 22, Willamette 8Marshfield 26, Churchill 6North Bend 24, Springfield °Week 6Willamette 27, Sheldon 22North Eugene 28, Churchill 6South Eugene 28, Thurston 21Marshfield 20, Springfield °North Bend 34, Cottage Grove 2110. Use the scores to make league standings for each of the weeks of theseason. Include the number of games won, number of games lost and thepercent of games won expressed as a 3-place decimal like .750.11. Were the scores of the first-place team all above the mode? median?mean?419

.j>.I\.)o"U (f)~'"03 c

1. A. What part ofthe body sustainedmoreinjuries thanany otherpart?B. What percent ofthe total wasit?C. Of the 1,169injuries howmany were kneeinjuries?2. The leg accountedfor what percent ofthe total NFL injuriesin 1974?Is this more orless than half?Perm'ission to use granted by Stanford Research Institute3. According to this study, of all injuries counted in 1974, 15.4% were reoccurencesof previous injuries.What % of the total were first time football injuries?4. What percent of all injuries occured above the shoulder?Of the 1,169 injuries how many does this account for?5. The survey showed younger players received more injuries than their older teammatesand the eight teams at the top of the league received fewer injuries than the bottomeight teams. Can you give some reasons for this outcome?421

FLICHT"rTHE"/,E88(In most classes a number ofstudents know how to make paperairplanes.They are usuallyready to test their planes in acontest.The Great International Paper Airplane Book by Jerry Mander,George Dipple andHoward Gossage offers many designs for paper airplanes for those students who do nothave favorite designs.(Contests can be held to measure the distance flown or the duration of the flight.Flights can be scored by assigning one hundred points to the best performance inany category and the remaining flights scored on what perce~'they are of the bestperformance.For example if the longest flight is 22.0 metres, it would score 100 points.Aflight of 16 metres would be scored as follows:1622 x 100 = .727 x 100 = 72.7 or about 73 pointsDecathlon.For an activity simulating a decathlon and using paper airplanes, see Classroom(422

To determine the score in shuffleboard find the sum of all puckscompletely contained in a region. Any puck that is touching a linedoes not score.1. A. Is it possible to score exactly 40 points with five pucks?If so, where does each puck lie?B. Is it possible to score exactly 40 points with four pucks?If so, where does each puck lie?C. Is it possible to score exactly 40 points with three pucks?If so, how?2. What is the greatest total you can score using four pucks?T~atis the smallest positive total you can scorewith four pucks?3. What would be the total for each of the following?3.:10 01.10 OFF4. Indicate for each of the scores 10 through 20 which are possibleto score with four pucks and which are impossible. For thosethat are possible give the individual puck placement.10 - 1 on the 10 14 - 18 -Il- lS - 19 -12 - 16 - 20 -13- 17 -10 OFF423

Vet Moyer in the waySullivan slates bidfor boxing's bigtimeBy NEIL CAWOODOf the Register-GuardThe curtain goes up on Eu·gene's biggest fight card in 10years tonight when EvergreenSports Inc. unveils a five~fightpro g ram featuring middle~wei g h t 5 Denny Moyer andJohn L. Sullivan at the LaneC 0 u n t y Fairgrounds' indoorarena.A crowd of approximately2,000 is expected to watch Sullivan'sbreak·through into box·ing's bigtime.Sullivan, winner in all of his20 professional bouts, will betesting his skills against one ofthe best and most experiencedin the middleweight division.Sullivan's progress has beenobserved first·hand in Eugene,from the time he fought hissecond fight on March 17,1971, through the time hefought No. IS this past February.A VICTORY against the seasoned33·year-old Moyer, winnerof 105 out of 137 ringbattles, would put Sullivan inline for ranking in the top 10of either the middleweight(160-pound limit) or juniormiddle-weight (154) divisions."Sure," agreed Sullivan, "awin here should give me a bigjump internationally. I'm unknown"internationally and ac~tually it's just the West Coastwhere I'm known."''It would be a big boost andthat's what it's ~1l about ­just keep eli m bin g thosestairs."Sullivan will not only beglVlllg up 14 years of experiencebut he'll be going intotonight's main event aboutthree pounds lighter than Moyer.424John L. Sullivan26157 pounds5'8"6SY,"3S%"39%,"13y,"11%"6%"11Y2"30"20%,"14%"9"Tale of the tapeAgeWeightHeightReachChest (Normal)Chest (Expanded)BicepsForearmWristFistWaistThighCalfAnkleMoyer33160 pounds5' 8~"69 1 4"39"40%"14Ys"11%"7"11%,"30Y,"20Y,"14y:!"8%"Newspapers and magazinescan provide thebasis for ~ mathematicsand sports activity.Students could collectarticles, write questionsabout the articlesand pass themaround the class.For the article tothe left, comparisonsof the two men's "statistics"could be made.a) Who is heavier?b) Whose measurementsare--on theaverage--greater?c) gow important arethe physicalaspects versusrecord of wins, orage?d) How many inchesdoes each man'schest increase bywhen expanded?A more involved projectwould be to comparetwo football orbasketball teams. Usetheir weights,heights, ages orrecords to predict awinner.(((

famou, f)lack Ameflcan,PuzzleThough only five feet five inches tall, this boxer is the only fighter in the history ofboxing to have held three world titles at the same time - the featherweight, welterweight, andlightweight titles. In 1954 he was elected to Boxing's Hall of Fame.5,382 7,488 53,572 16,037 16,43222,904 16,037 45,828 28,576 32,760 16,037 39,690 53,572 24,843...............................Complete the problems below. Use the code letters above._.........................5each problem to find the name of the famous person.A.409x 5 6B.706x 7 8C.304x 4 9D.208x 3 7E.208x 3 6F.306x 8 7G.507x 4 9H.207x 2 6I.705x 9 8J.602x 8 7l.906x 8 9M.603x 7 6N.908x 5 9O.405x 9 8P.704x 6 7R.203x 7 9S.304x 9 4T.504x 6 5U.609x 3 8W.206x 2 9Y.208x 7 9SOURCE: Famous Black Americans Number Puzzles, Book BPermission to use granted by Creative Teaching Associates 425

IfnMOUf nTIILETEf PUZlLE I(Born in 1927 in the town of Silver, South Carolina, this person showed an interest in all sports. \.She developed into an outstanding tennis player and in 1957 became the first Black American womanto win the United States National Singles Championship. She also won the Wimbledon Women'sSingles Championship. She won both titles again in 1958 and then retired from amateur tennis. In1957 she became the first Black woman to be voted the Athlete of the Year. After retiring fromtennis she became a golf professional and a night club singer.8:09 11 :01 9:09 11 :06 11:00 8:09 11:03 11: 12 11: 13 11 :35 12: 12 9:10A B C D E4:2 7 8:3 6 9: 1 7 2:5 5 7: 1 2+ 3:4 2 + 2:3 7 + 1: 4 8 + 6:2 1 + 3:4 8F5:2 4+ 6:4 7G8:0 5+ 2:5 8H3:2 7+ 7:3 96:5 5+ 4:1 7J1 : 1 9+ 7:5 3(KLMNo4:4 3+ 5:3 88:3 6+ 2:2 51 0:2 8+ : 4 72:2 1+ 6:4 94: 3 3+ 7: 3 9426P R S T U1 1 : 3 8 4:4 1 7:4 9 5:5 1 4:0 9+ :3 6 + 5:4 5 + 3:4 6 + 3: 1 8 + 5:5 7SOURCE:Famous Athletes Number Puzzles, Book APermission to use granted by Creative Teaching Associates(

BACK.COURTThe diagram to the leftis a scale drawing of atennis court using ascale of 1 cm:l m.1) Find the length,width, perimeter andarea of a back court.lengthwidthperimeterareaRIGI·\ISERVICECOURTALLEYLEFTSER\)ICI;:COUR,BACK.COURT4) Use the informationin exercises 1, 2 and3 to find the length,width, perimeter andarea of the entiretennis court.lengthwidthperimeterareaIDEA FROM:MathmSports,CILAMPPermission to use granted by Central Iowa Low·Achiever Mathematics Project427

Have a student bring a 5 or 10 speed bicycle to class. Turn the bike upside down sothat the gears can be shifted. PlIt a piece of tape on the rear wheel of the bicycle.Have the students count the teeth in -, Eeach gear and record in Table 1- (The ..Bnumber is not standard. The front gears rCvary from 52 to 39 teeth and the rearrDAX.-.,;'gears from 34 to 14.):::=:::sa~Write the gear ratios and simplify.(/~Record in Table 2.The following activities afor student investigation:TABLE1. Select a simplegear ratio, for example13 to 4, and set thegears to correspond.Check the gear ratio byslowly turning thepedals. The pedalsshould turn four timesand the wheel thirteen.(Hold the rear tirelightly to aid incounting the turns ofthe wheel.) Checksome other gear ratiosby counting pedal andrear wheel turns.re suggestedTABU: 1Geo.vXyAB2. Select a back gear and use the smallshift to the large front gear. Continuechange do you notice in the back wheel?gear ratios?Numb~of' teetc Xto r.DEGeo.v Ratio of Simplifiednum b erRa.tio of teeth Teeth R.o..+ioXtoAXto BX+oC)( +0 DYtoAYto BYto CYi-o Dyto Efront gear. Turn the pedals slowly andturning the pedals at the same rate. WhatCan you explain? What are the corresponding2((3. Move the gearshifts so the chain is on the smallest back and front gears. Turnthe pedals at a constant rate. Shift only the back gear so that the chain travelsfrom the smallest to the largest gear wheel. What change occurs in the back wheel?Can you explain? What are the corresponding gear ratios?4. If the pedals were turned at a constant rate, which ratio would cause the backwheel to turn the fastest? Order the simplified gear ratios from largest to smallest.Students could use a calculator to change each ratio to a decimal and then order thedecimals.5. In riding the bicycle, which gear setting is the easiest to pedal? the mostdifficult? Experiment on the playground. Which gear setting allows you to travelthe farthest for one turn of the pedal? Devise a method for checking your prediction.4286. In a cross-country bicycle race which gear setting would be best for going up asteep hill? (7. In a cross-country bicycle race which gear setting would be best for the sprint tothe finish line?

The table below gives the number of boating accidents reported to the U.S.in 1973. Use the table to answer these questions.Coast Guard1) a.b.c.2) a.b.c.What percent of the accidents occurred in non-tidal waters?What percent of the fatalities occurred in non-tidal waters?What percent of the injuries occurred in non-tidal waters?What percent of the accidents occurred when visibility was poor?What percent of the fatalities occurred when visibility was poor?What percent of the injuries occurred when visibility was poor?3) What percent of the accidents occurred whena) the water was calm?b) the weather was clear?c) the wind was light?d) the visibility was good?4) These statistics seem to imply that the worst time to go boating is when the wateris calm, the weather is clear, the wind is light and the visibility is good. Whatdo you think?WEATHER AND WATER CONDITIONSIII~.~w"i~./ 1973~w ;: ~..1..1> wJ..,W..l .., ii"'} 1790

Use the chart to answer these questions. Note that the attendance figures aregiven in units of 1000 people. So, 17,659 means 17,659 x 1000 or 17,659,000.(a) lihich year shows the fewest number of professional boxers?b) Did more people attend college or professional football games?c) In 1973 how many people attendedmajor league baseball games?college football games?d) Did attendance always increase in baseball games?football?basketball?e) Which attendance records more than doubled between 1960 and 1973?f) Baseball attendancebasketball?showed what per~ent of increase between 1950 and 1973?professional football?g) Baseball's attendance of about 30 million is what percent of the U.S. population of210 million?h) Basketball's attendance of about 7 million is what fracti?nal part of the U.S. populationof 210 million?what percent?i) Estimate basketball's attendance for 1974. (j) Make a graph to show the "ise in attendance for baseball, basketball and collegefootball for the years 1950 to 1973.RECREATION. Selected Recreational Activities: 1950 to 1973Activity Unit 1950 1960 1965 1970 1971 1972 1973Baseball, major leagues:Attendance 1,000 17,659 20,261 22,806 29,000 29,544 27,330 30,467Regular season 1.000 17,463 19,911 22,442 28,747 29,193 26,967 30,109National league 1,000 8,321 10,685 13,581 16,662 17,324 15,529 16,675American league 1,000 9,142 9,227 8,861 12,085 11,869 11,438 13,434World series 1,000 196 350 364 253 351 363 358Basketball, professional, attendance:National Basketball Assoc. 1,000 1,986 2,750 5,1L~7 6,195 6,634 6,834American Basketball Assoc. :Regular season 1,000 1,753 2,230 2,437 2,400Playoffs 1,000 213 299 360 364Football:College, National CollegiateAthletic. Association:Teams Number 674 620 616 617 618 620 630Attendance 1,000 18,962 20,403 24,683 29,466 30,455 30,829 31,283Professional, Nat'l Football League:Attendance 1.000 2,115 4.153 6,546 9,991 10,560 10,929 11,257Regular season 1,000 1,978 4,054 6,416 9,533 10,076 10,446 10,731Championship games 1,000 137 99 130 458 484 483 526Boxing, professional matches: \Boxers Number 3,940 2.920 2,202 5,071 5,783 6,136 7,384Receipts. gross $1,000 3,800 5,902 8,264 10,642 10,237 11,847 12,634,(430

••••• ~~..... _ ...._ •• _ •• _ ••••Charts or tables enable a person to find facts rapidly because the information iswell-organized and kept as brief as possible. Use the information in these charts toanswer these questions.PARTICIPATION IN SELECTED OUTDOOR RECREATlON ACTIVITIES: 1972ACTIVITYCamping in remote or wilderness areash_~ _Camping in developed ClUllp grounds.•m ••••~!~gt~~~~~":::::::::::::::::::::::::::::::::::Riding motorcycles off the road'_h_'~_m.•_Hiking with a pack, mountain/rock climbing.Nature walksnn_._. .•n_._•.".,•. _.• __Walking ror pleasure._no-h•• ___Bicycling••.••••••• _._Horseback rlding__ h u''' ' ._..._hh__•••_h. __ Wh_'Water skiing.. . _n '_" __ h n_'n_hUn~~~~gboating::::::::::::::::::::::::::::::::Outdoor pool sw1mmlng.m __ •• m.~.~.. "'Other swimming outdoors•.m u .Oolfn._••• ~•••.••••• ~••••' .••••••••....••••••Tennis••••...•••.•••••....•••• ~••••.••• ~••••.Playing other outdoor games or sportsm •••••Going to outdoor sports evenl:s~•.••• uu~••••Visiting tOOS, lairs, amusement parks..•d ••••~~~~1~i1~~·::::::::::=::::=:::=·::::::::::::::Driving for pleasure.•.•••••• ~•.••••• ~~...••••Snow skiing ~.~'•. '_.~•• ~.'_ .•• ~.~.SnOWDlobl1ing•• nOther winter sports•• ~u nnn~~••• nU~h.nParticipants(millions)7.717.522.238.'7.'8.'26.7...,16.78.78.5~123.328.563.87.78.'35.'18.938.759.874A...,ParticIpantsas percentageof totalpopulation tData from United States Statistical Abstract, 1974.7.'7.'24.8,111424,5173410553151834552'21224374734,16a) What is the most popularactivity? _....c.-'-'-'-'''''''.;c,'".""" _b)c)d)e)f)g)The second most popular?What fraction of the U.S. populationuses picnicking as anactivity?fishingas an activity?Do more people camp in campgroundsor wilderness areas?Do more people hunt or fish?snow ski or water'ski?What fraction of the populationbicycles?Do twice as many people bicycleas play golf?h) Are golf, tennis and fishing allequally popular?No. 344. FISHING AND HUNTING LICENSES-NuMBER AND COST: 1950 TO 1972[For years ending June 30. l'rlor to 1960, fishing and huntlng licenses exclude Alaska and HawaiI. See also Historical a)Statistics Colonial Times to 1957 series H 523.524),ITE~{ 1950 1955 1960 1965 1968 1969 1970 1971 1972------------------Field ngIicense&S ~ Sales1..mlllions.. 15.3 18.9 23.3 25.0 28.8 29.9 31.1 32.4 33.0Resident•••..••".._.mUllons.. 13.3 16.2 20.2 21.6 24.9 25.7 26.8 27.8 23. ,Nonresident.. ~ ~ _••mUlions.. 2.0 2.6 3.1 3.' 3.' '.2 '.3 .., '.8Cost to anglers,,"nn.ffill. doL 34 .0 52 63 80 88 91 100 107HunUnglkenaesl Sales l.mlllions.• 12.6 14,2 18.4 19.4 20.9 21.6 22.2 22.9 22.2Resident-••••.•."...millions._ 12.4 13.9 17.8 18.5 19,9 20.6 21.0 21.6 21.0Nonresldent.•••....milJions_. .2 .,.9 .9 1.0 1.2 1. , 1.2Cost to huntersuhu.mll. doL. 38 43 51 75 89 96 102 109 107Federal duck. e.amps sold.•1, 000•. 1,955 2,185 1,629 1,566 1,934 1,839 2,072 2,420 2,442I Prior to 1960, paid license holders; for definition, see footnote 1, table 346.Source: U.S. Fish and Wlldllfe ServIce, FederalAid in Fish a1ld Wildlife Restoration, annual, and unpUblished data.Data from United States Statistical Abstract, 1974.'Id) Sales in 1970 rose what percent over the 1950 sales for fishing?e) Sales in 1972 rose what percent over 19501'.'':/"Why would there probablybe an increase during1960?b) During all the years?c) Were more fishing orhunting licenses soldover the years?f) Which sport doubled in number of licenses sold using the years 1950 and 1972?g) The sales for both sports from 1950 to 1972 represent what percent of increase?h) The cost of fishing licenses rose from approximately $2 tohunting rose from $3 to __~~ _The cost of431

(((

C£OJSARVat bat. In baseball the number oftimes a player is up to bat, exceptno time at bat is recorded when theplayer (1) hits a sacrifice bunt orsacrifice fly, (2) is awarded firstbase on four balls, (3) is hit by apitched ball, or (4) is awarded firstbase because of interference or obstruction.base hit. In baseball when a battersafely reaches first base on a fairball. A base hit is not recordedwhen the batter safely reaches firstbase because of a fielding error orbecause the other team tried to putout another base runner.batting average. In baseball a decimalnumber for each batter found bydividing the number of base hits bythe number of times at bat.bowling ball. Ball that weighs between10 and 16 pounds; used toknock down pins.bowling pins.a triangulartry to knockTen woodenarrangementdown.objects inthat youcalorie. 1. One calorie is theamount of heat needed to raise thetemperature of 1 gram of water byOne Celsius degree (also calledsmall calorie). Symbol: cal2. (usually spelled with a capitalC) 1000 calories. (also calledlarge calorie). Symbol: kcal; Caldash. In track a short running event.1. 100-yard dash (to be replaced bythe 100Imetre dash). 2. 220-yarddash; 8 of a mile lto be replacedby the 200 metre dash). 3. 440­Iyard dash; ~ of a mile (to be replacedby the 400-metre dash).decathlon. A track competition consistingof 10 events: 100-metredash, long jump, shot put, high jump,400-metre dash, 110-metre hurdles,discus, pole vault, javelin, andl500-metre run.discus. A disc-shaped object with amass of 2 kilograms (about 4.4 pounds)thrown in a track meet. The discusfor women has a mass of 1 kilogram(about 2.2 pounds).error. In baseball when a playerfumbles a baIlor makes a wild throwthat allows a runner to advance Oneor more bases.gear ratio. Comparison of the numberof teeth on a front gear to the numberof teeth on a back gear of abicycle.home run distance. The distance fromhome plate to the outfield fence.This varies from stadium to stadiumand varies within a stadium if theplaying field is irregular in shape.infield. The general area of a baseballfield enclosed by the three bases andhome plate, and within a 95-foot radiusof the pitching rubber.javelin. A spear-like object 260centimetres long with a mass of 800grams thrown in a track meet. Thejavelin for women has a mass of 600grams.mean (arithmetic). The value found byadding all the numerical items of agroup and dividing by the number ofitems in the group. Often called theaverage.median. In a group of numbers arrangedin order the number that is in themiddle.mode. The number that occurs mostoften in a group of numbers.muscle fatigue. When muscles gettired from doing work.Olympic games. An international competitionheld every 4 years.atb-oly433

SPORTS GLOSSARY ope-wor 2open frame. In bowling a frame wheresome pins are still standing after2 rolls of a bowling ball.outfield. The part of a baseballfield in fair territory that is notthe infield.penthalon. An athletic competition consistingof 5 events: 300-metre freestyleswim, 4000-metre cross-countryrun, SOOO-metre 30-jump equestriansteeple chase, epee fencing and targetshooting at 25 metres.pulse.Number of heartbeats per minute.range. The difference between thehighest and lowest values in a groupof numbers.stagger. 1. 1 turn--the headstart (distance given to runners in a trackmeet where runners must stay intheir respective lanes for one curveof the track. 2. 2 turn--the headstartdistance given to runners in atrack meet where runners must stayin their respective lanes for twocurves of the track.strike. 1. A frame in which all tenpins are knocked down with 1 roll ofa bowling ball. 2. When a batterswings and misses a pitched ball inbaseball or softball. 3. When apitched ball is in the strike zoneand a batter does not swing at it.4. A foul ball when the batter hasless than two strikes.world record.In track the fastestrun. In track a lo~g running event.time or farthest distance officially1. 880 yard run; - of a mile (to be recorded in the world for an event.replaced by the 800-metre run). 2.Mile run (to be replaced by the 1500­metre run). 3. 3-mile run (to bereplaced by the SOOO-metre run). (sacrifice. 1. bunt--with less thantwo out~ the batter advances one ormore base runners with a bunt and isput out at first base. 2. fly--withless than two outs, the batter hitsa fly ball that is caught and a baserunner scores after the catch.shot put. In track a heavy metalball (7.3 kilograms in college)thrown in a track meet. The shotput for women has a mass of 4 kilograms.slugging percentage. In baseball adecimal number for each batterfound by dividing the number oftimes at bat into the total basesof all safe hits.spare. A frame in which all ten pinsare knocked down with 2 rolls of abowling ball.434

ANNOTATE" BZKZOCIl4lPWThe following is a list of sources used in the development of this resource.It is not a comprehensive listing of materials available. In some cases, goodsources have not been included simply because the project did not receive permissionto use the publisher's materials or a fee requirement prohibited its use by theproject.ACTIVITIES IN MATHEMATICS, Second Course. Johnson, Donovan A; Hansen, Viggo P.;Peterson, Wayne H.; Rudnick, Jesse; Cleveland, Ray; and Bolster, L. Carey.Glenview, Illinois: Scott, Foresman and Company, 1973. (Scott, Foresmanand Company, 1900 East Lake Ave., Chicago, IL 60025)384 pp; cloth; textbook; b/wThis book is an elementary text in which many of the conceptsare developed through use of manipulatives and activities.ACTIVITIES IN MATHEMATICS, Second Course. Johnson, Donovan A.; Hansen, Viggo P.;Peterson, Wayne Ho; Rudnick, Jesse A.; Cleveland, Ray; and Bolster, L. Carey.Glenview, Illinois: Scott, Foresman, and Company, 1973. (Scott, Foresmanand Company, 1900 East Lake Ave., Glenview, IL 60025)432 pp; cloth; teacher's guide; colorA collection of 85 student activities covering four topics:graphs, statistics, proportions and geometry are contained inthis book. Each of the four sections can be purchased separatelyin paperback format for student use.ALL AROUND YOU. Bureau of Land Management. Washington, D.C.:Printing Office, 1971. (U.S. Government Printing Office,Documents, Washington, D.C. 20402)126 pp; paper; workbookU.s. GovernmentDiv. of PublicThis collection of student worksheets and teacher's pages isoriented toward middle school environmental education and containsa few activities which could lead to mathematical work.THE AMBIDEXTROUS UNIVERSE. Gardner, Martin. New York: Basic Books, Inc., 1964.(Basic Books, Inc., 10 E. 53rd St., New York, NY 10022)cloth;teacher reference; b/wThis book is full of light and humorous stories which relyheavily upon an understanding of science for explanation.435

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYARITHMETICAL EXCURSIONS: AN ENRICHMENT OF ELEMENTARY MATH. Bowers, Henry andBowers, Joan E. New York: Dover Publications, Inc., 1961. (Dover Publications,Inc., 180 Varick St., New York, NY 10014)320 pp; paper; b/w; teacher referenceThis is a book of enrichment topics in mathematics ranging fromcounting through the arithmetic operations to figurate, perfect andamicable numbers and then on to mysteries and folklore of numbers.Exercises and answers are provided for each of the 27 topics.ASK ANDY. Los Angeles: Los Angeles Times SyndicateASK ANDY is a syndicated column in newspapers. Childrensubmit science-related questions. Andy then responds innon-technical language. Each child he responds to is senta 20-volume encyclopedia."The basis of music--mathematics." Lawlis, Frank. THE MATHEMATICS TEACHER, Vol. LX,No.6 (October, 1967), pp. 593-596.This is an easy-to-read discussionapplied to rhythm, harmony and melody.ideas for "Cube Compositions."of how mathematics has beenThis article provided the(BETTER ENVIRONMENT BOOKLETS. Soil Conservation Society of America. (Soil ConservationSociety of America, 7515 N.E. Ankeny Rd., Ankeny, IA 50021)16 pp. each; paper; student referenceThis series of comic books have environmental themes, withoccasional numerical data which could be the basis for mathematicalquestions. Titles are "Working Together for a LivableLand" (1970); "Making a Home for wildlife on the Land"; "TheStory of LandI',. "Plants, How They Improve Our Environment!!:;"Help Keep Our Land Beautiful"; "The Wonder of Water" (all1971); and "Food and the Land" (1972).(THE BICYCLE VS. THEAgency, 1974.D.C. 20460)ENERGY CRISIS. Washington, D.C.: U.S. Environmental Protection(U.S. Environmental Protection Agency, 401 M st., S.W., WashingtonThis pamphlet describes the advantages of bicycling in termsof its contribution to energy conservation.BOATING STATISTICS 1973. Washington, D.C.: U.S. Department of Transportation,1974. (U.S. Department of Transportation, 400 Seventh St., S.W., WashingtonD.C. 20590)44 pp; paper; teacher referencE; b/wThis is a compilation of statistics about boating and boatingaccidents for 1973.(436

BIBLIOGRAPHYMATHEMATICS IN SCIENCE AND SOCIETYBUILD-IT-YOURSELF SCIENCE LABORATORY.Doub.leday & Company, Inc., 1963.Ave., New York, NY 10017)340 pp; cloth; teacher reference; b/wBarrett, Raymond. Garden City, New York:(Doubleday and Company, Inc., 245 ParkThis book contains a collection of science activities whichcan be set up and performed using homemade, inexpensive equipment.CHARTING THE UNIVERSE, Book 1. (The University of Illinois Astronomy ProgramSeries) Elementary-School Science Project. Co-Directed by Atkin, Myron J. andWyatt, Stanley P. Jr. New York: Harper and Row, Publishers, 1969. (Harperand Row Publishers, 10 E. 53rd St., New York, NY 10022)87 pp; paper; textbookThe University of Illinois Program is a series or six books thatintroduces astronomy to upper elementary and junior high students.The major concepts in each book are explored through numerous studentactivities as well as through the development of models thatexplain astronomical phenomena. The series is an excellent-resourcefor applications of mathematics to astronomy.CHARTING THE UNIVERSE, Book 1, Guidebook. (The University of Illinois AstronomyProgram Series) Elementary-School Science Project. Co-Directed hy Atkin,Myron J. and Wyatt, Stanley P. Jr. New York: Harper and Row, Publishers, 1969.(Harper and Row Publishers, 10 E. 53rd St., New York, NY 10022)72 pp; paper; teacher's guideThe university of Illinois Program is a series of six books thatintroduces astronomy to upper elementary and junior high students.The major concepts in each book are explored through numerous studentactivities as well as through the development of models that explainastronomical phenomena. The series is an excellent resourcefor applications of mathematics to astronomy.CHILDREN AND ANCESTORS, Action Biology Series. Weinberg, Stanley L. and Stoltze,Herbert J. Boston: Allyn and Bacon, Inc., 1974. (Allyn and Bacon, Inc.,470 Atlantic Ave., Boston, MA 02210)64 PPi paper; workbook; color; medium reading levelThis workbook contains fifteen activity-oriented lessonsdealing with topics on genetics.CLEAN AIR AND YOUR CAR. Washington, D.C.: U.S. Environmental Protection Agency,n.d. (United States Environmental Protection Agency, Office of Public Affairs,Washington, D.C. 20460)437

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYFROM HERE, WHERE? A Sourcebook in Space Oriented Mathematics for Secondary Levels.Washington, D.C.: Superintendent of Documents, U.S. Government Printing Office,1965. (U.S. Government Printing Office, Div. of Public Documents, Washington,D.C. 20402)192 pp; paper; student reference; high reading levelThis is a collection of applications of mathematics in spacescience that is designed to supplement the material usually studiedin secondary school mathematics.(FUNDAMENTAL ASTRONOMY: SOLAR SYSTEM AND BEYOND.John Wiley and Sons, 1974. (John Wiley andNew York, NY 10016)476 pp; cloth; textbookCole, Franklyn W. New York:Sons, Inc., 605 Third Ave.,The book is a text for a beginning college level course inastronomy that will provide adequate information for those desiringa knowledge of fundamental astronomy.A GALAXY OF GAMES FOR THE MUSIC CLASS. Athey, Margaret and Hotchkiss, Gwen. WestNyack, New York: Parker Publishing Company, Inc., 1975. (Prentice-Hall, Inc.,Englewood Cliffs, NJ 07632)215 pp; cloth; game bookThis book contains a collection of 241 games and teacher aidsfor teaching rhythmic response, melody writing, music notationand ear training. Each game is rated K-8.(GENETIC CONTINUITY. Glass, Bentley. Biological Sciences Curriculum Study.Lexington, Massachusetts: D.C. Heath and Company, 1968. (D.C. Heath andCompany, 125 Spring St., Lexington, MA 02173)154 pp; paper; workbook; b/w; high reading levelThis is a workbook designed to allow students to discover variousideas dealing with genetics and inherited traits.GEOMETRY IN MODULES, Book B. Lange, Muriel. Menlo Park, California: Addison­Wesley Publishing Company, Inc., 1975, pp. 103-110. (Addison-Wesley PublishingCompany, Inc., 2725 Sand Hill Rd., Menlo Park, CA 94025)124 pp; paper; textbook; b/w; medium reading levelThese pages contain ideas on how changes in linear dimensionsaffect changes in surface area and volume.(442

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYGIANT GOLDEN BOOK OF MATHEMATICS-Exploring the World of Numbers and Space. Adler,Irving. New York: Golden Press, Inc., 1958, 1960. (Western Publishing, Inc.,1220 Mound Ave., Racine, WI 53404)56 pp; cloth; student reference; colorDesigned to interest young people, this is a book of mathematicalideas.THE GOLDEN MEASURE. (Reprint Series) Schaaf, William L., Ed. Palo Alto: SchoolMathematics Study Group, 1967. (Distributed by A.C. Vroman, Inc., 2085 E.Foothill Blvd., Pasadena, CA 91109)46 pp; paper; teacher reference; b/wThis teacher reference book reprints several articles dealingwith. the golden ratio and its occurrence in nature, art, and geometry.GREAT CURRENTS OF MATHEMATICAL THOUGHT, Volume II: MATHEMATICS IN THE ARTS ANDSCIENCES. "Mathematics and music." Edited by LeLionnais, F. New York:Dover Publications, Inc., 1971, pp. 189-196. (Dover Publications, Inc.,180 Varick St., New York, NY 10014)teacher referenceA discussion of the mathematical basis of musical scales and thevarious number relations in music are given here.GROVE'S DICTIONARY OF MUSIC AND MUSICIANS, Volumes I-X. Blom, Eric, ed. New York:Macmillan and Company, LTD, 1954.cloth; teacher referenceThese ten large volumes contain an extensive coverage of musicalterms, composers, compositions, style and history.GUINNESS BOOK OF WORLD RECORDS. McWhirter, Norris and McWhirter, Ross.Sterling Publishing Co., Inc., 1975. (Sterling Publishing Company,419 Park Ave. South, New York, NY 10016)688 pp; cloth; student reference; b/w; medium reading levelNew York:Inc. ,HARVARD DICTIONARY OF MUSIC.University Press, 1966.MA 02138)Apel, Willi. Cambridge, Massachusetts: Harvard(Harvard University Press, 79 Garden St., Cambridge,cloth; teacher referenceThis is a one-volume dictionary which includes history, backgroundand use of musical terms.443

BIBLIOGRAPHYMATHEMATICS IN SCIENCE AND SOCIETY(ENERGY CRISES IN PERSPECTIVE. Fisher, John. New York: John Wiley and Sons, Inc'.,1974. (John Wiley and Sons, Inc., 605 Third Ave., New York, NY 10016)196 pp; cloth; teacher reference; b/wThis book contains many tables and graphs about energy use andsources.ENERGY PRIMER. Menlo Park, California, Portola Institute, 1974. (~he 1967 editionhas been published by Dell Books, Dell Publishing Company, Inc" 1 Dag Hammar~skjold Plaza, 245 E. 47th St., New York, NY 10017)200 pp; paper; teacher referenceThis book gives a survey of small-scale energy production by solar,water I wind and biofuels. It also reviews several books.ENVIRONMENTAL EDUCATION IN THE ELEMENTARY SCHOOL. Sale, Larry and Lee, Ernest.New York: Holt, Rinehart and Winston, Publishers, 1972. (Holt, Rinehartand Winston, Publishers, 383 Madison Ave., New York, NY 10017)203 PPi paper; teacher referenceThis book provides teacher background and some ideas for environ~mental education in the classroom. The activities are not describedwith mathematics in mind, but there are lots of numerical data whichmight suggest ideas.(ENVIRONMENTAL SCIENCE, PROBING THE NATURAL WORLD, LEVEL III. Intermediate ScienceCurriculum Study. Morristown, New Jersey: Silver Burdett General LearningCorporation, 1972. (Silver Burdett General Learning Corporation, 250 Jamesst., Morristown, NJ 07960)146 pp; textbook; colorThis is an excellent collection of environmental activities,with many mathematics-related questions built in.ESSENCE I (revised). Menlo Park, California: Addison-Wesley PUblishing Company,Inc., 1971. (Addison-Wesley Publishing Company, Inc., 2725 Sand Hill Rd.,Menlo Park, CA 94025)activity cards; teacher's guidei b/wThis series of "transdisciplinary," "open," student-centere¢lactivities are designed to build an atmosphere of trust. Someinvolve applications of mathematics.(440

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYFAMOUS ATHLETES NUMBER PUZZLES.Teaching Associates, 1974.Fresno, CA 93727)Silvani, Harold. Fresno, California: Creative(Creative Teaching Associates, P.O. Box 7714,48 pp, paper, puzzle book; b/w; low reading levelThis is a book which provides drill in addition, subtraction,multiplication, and division of whole numbers and fractions, andalso includes addition and subtraction of time and changing fractionsto decimals and percents. A format of puzzles wi th tI FamousAthletes" as answers is used.FAMOUS BLACK AMERICANS NUMBER PUZZLES. Ecklund, Larry. Fresno, California: CreativeTeaching Associates, 1975. (Creative Teaching Associates, P.O. Box 7714, Fresno,CA 93727)56 pp; paper, puzzle book, b/wThis is the second of two books covering basic skills in addition,subtraction, multiplication, and division using a format of puzzleswith "Famous Black Americans" as answers.FIBONACCI AND LUCAS NUMBERS. Hoggatt, Verner E., Jr. Boston: Houghton MifflinCompany, 1969. (Houghton Mifflin Company, 1 Beacon St., Boston, MA 02108)92 pp; paper, teacher reference; b/wThe book offers an introduction to some of the properties ofFibonacci and Lucas numbers.FRACTION BARS.Colorado:2121, Fortactivity kitBennett, Albert B. Jr.Scott Resources, Inc.,Collins, CO 80522)and Davidson, Patricia S. Fort Collins,1973. (Scott Resources, Inc., P.O. BoxThis kit contains a program of games, activities, manipulativematerials, objectives, workbooks, and tests for teaching fractions.A model for fractions with denominators of 2, 3, 4, 6 and 12 is usedthroughout this very excellent kit.FRACTION FACTORY. Minneapolis, Minnesota: The Math Group, Inc., 1973. (TheMath Group, 396 East 79th St., Minneapolis, MN 55420)21 activity cards; b/wTwenty-one cards presenting situations where fractions are needed forsolution of a problem make up this set.441

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETY("Computer analysis of musio forms." 4adner, Robert .Jr. MUSIC JOURNAL, Vol. 26(October, 1968), pp. 33, 58.A short disoussion of the praotioality of analyzing musiovia computer in order to determine who the composer was is givenhere."Computer musio." Hiller, Lejaren A. Jr. SCIENTIFIC AMERICAN, Vol. 201 (December,1959), pp. 109-120.The author of this article asks,"Can computers be used to composea symphony?u, An affirmative answer is given by explaining what canand has been done. Examples are shown from ILLIAC SUITE FOR STRINGQUARTET whioh was oomposed by ILLIAC, the high-speed digital oomputerof the University of Illinois."Computers and future music." Mathews, M.J.; Moore, F ..'R.; and Risset, J.e.SCIENCE, Vol. 183, No. 4122 (January 25, 1974), pp. 263-268.This is a discussion of computers and music. Groove (generated£eal-time ~perations ~n ~oltage-oontrolled~quipment) is disoussedin some detail."The oonoept of unity of eleotronio musio." Stockhausen, Karlheinz. PERSPECTIVESOF NEW MUSIC, Vol. 1 (Fall, 1962), pp. 39-48. (This is a technical article on the author's experimentationwith electronic music.COSMIC VIEW:Company,10019)THE UNIVERSEInc or 1957.IN FORTY JUMPS. Boeke, Kees. New York: John Day(John Day Company., Inc., 666 Fifth Ave., New York, NY"Counterpoint as an equivalenoe relation." Delman, Morton. THE MATHEMATICS TEACHER,LX, No.2 (February, 1967), pp. 137-138.This is a short artiole explaining that the use of oounterpoint,is an equivalence relation* The article references several jazzreoords in whioh students oan hear eaoh player improvising, yet itall blends together because counterpoint is "transitive. ItDEMONSTRATION EXPERIMENTS IN PHYSICS.Company, 1938. (McGraw-Hill BookYork, NY 10020)545 pp; cloth; teacher reference; b/wSutton, Richard. New York: McGraw-Hill BookCompany, 1221 Avenue of the Americas, NewThis is a oolleotion of ideas and demonstrations. Althoughit was published in 1938, it's an exoellent resouroe. Its contentsare still usable, and many with a minimum of equipment.(438

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYDIMENSIONS AND MOTIONS OF THE EARTH . .. MEASURING TIME, SPACE AND MATTER, TeacherFolio-Investigation 5. (Time, Space and Matter ... Investigating the PhysicalWorld Series) Secondary School Science Project. Princeton, New Jersey: WebsterDivision, McGraw-Hill Book Company, 1969. (McGraw-Hill Book Company, 1221Avenueof the Americas, New York, NY 10020)75 pp; paper; teacher's guide; b/wIncluded in the measurement activities in the booklet are articlesin which the student measures the circumference of the earth and usesthe rotational motion of celestial objects as a basis for our units oftime.THE DIVINE PROPORTION. Huntley, H.E. New York: Dover Publications, Inc., 1970.(Dover Publications, Inc., 180 Varick St., New York, NY 10014)paper;teacher referenceMusic is referenced frequently in this collection of interestingideas related to the "golden section" of the Greeks.DOING THEIR THING, Action Biology Series. Weinberg, Stanley L. and Stoltze, HerbertJ. Boston: Allyn and Bacon, Inc., 1974. (Allyn and Bacon, Inc., 470 AtlanticAve., Boston, MA 02210)64 PPi paper; workbook; color; medium reading levelThis is a workbook of fifteen activity-oriented lessonsdealing with the senses and how we learn.THE EARTH AND HUMAN AFFAIRS. National Academy of Sciences. San Francisco: CanfieldPress, 1972. (Harper and Row Publishers, 10 E. 53rd St., New York, NY 10022)ECOLOGY, ENVIRONMENT, AND EDUCATION. Tanner, R. Thomas. Lincoln, Nebraska: ProfessionalEducators Publications, 1974. (Professional Educators Publishing, P.O.Box 80728, Lincoln, NB 68501)106 pp; paper; teacher referenceThis book describes some programs and materials and provides backgroundon issues in environmental education. ItWQuld be of most useto the teacher who wishes to explore environmental education extensively.THE ELECTRICAL PRODUCTION OF MUSIC. Douglas, Alan. New York: Philosophical Library,1957. (Philosophical Library, 14 E. 40th St., New York, NY 10016)"Electronic Music: its composition and performance .. " Moog, .Robert A. ELECTRONICSWORLD, Vol. 77, No.2 (February, 1967).THE ENCYCLOPEDIA OF SPORTS. Menke, Frank Grant. South Brunswick: A.S. Barnes,1975. (A.S. Barnes and Company, Inc., P.O. Box 421, Cranbury, NJ 08512)1125 pp; cloth; teacher reference; b/w439

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYHORNS, STRINGS AND HARMONY. Benade, Arthur H. New York: Anchor Books, Doubleday.and Company, Inc., 1960. (Doubleday and Company, Inc., 245 Park Ave., NewYork, NY 10017)paper;teacher referenceThis is a book giving a clear and comprehensive account of boththe scientific and the aesthetic nature of music.IDEAS AND INVESTIGATIONS IN SCIENCE-BIOLOGY. Wong, Harry K. and Dolmatz, Malvin S.Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1971. (Prentice-Hall, Inc.,Englewood Cliffs, NJ 07632)280 pp; cloth; textbook; b/w; medium reading levelFive ideas-~inquiry, evolution, genetics, homeostasis, andecology--are explored using activity--oriented lessons.INFORMATION PLEASE ALMANAC, ATLAS ANDDan Golenpaul Associates, 1974.New York, NY 10022)992 pp; paper; teacher reference; b/wYEARBOOK 1975. Golenpaul, Ann. New York:(Dan Golenpaul Associates, 502 Park Ave.,This book is a collection of national and world facts andevents.IN ORBIT, PROBING THE NATURAL WORLD, Level III. Intermediate ScienceStudy. Co-directed by David D. Redfield and William R. Snyder.New Jersey: Silver Burdett General Learning Corporation, 1972.Burdett General Learning Corporation, 250 James St., Morristown,III pp; paper; teacher referenceIN ORBIT is a science unit developed for students ages 11through 16 that uses a hands-on discovery approach to explorephenomena in the solar system.CurriculumMorristown,(SilverNJ 07960)INQUIRY INTO BIOLOGICAL SCIENCE, Teacher's Annotated Edition. Jacobson, Willard;Kleinman, Gladys; Hiack, Paul; Carr, Albert; and Sugarbaker, John. New York:American Book Company, 1969. (American Book Company, 450 W. 33rd St., NewYork,NY 10001)455 pp; cloth; textbook; color; high reading levelThis is a biology textbook organized around several conceptsof biology--namely science, the cell, plants and animals, thehuman organism, ecology, and continuity and change.((INTERACTION OF MAN & THE BIOSPHERE, Teacher's Edition. Abraham, Norman; Beidleman,Richard G.; Moore, John A.i Moores, Michael; and Utley, William J. InteractionScience Curriculum Project. Chicago: Rand McNally & Company, 1970. (RandMcNally and Company, P.O. Box 7600, Chicago, IL 60680)378 pp; paper; textbook; color; high reading levelThis is a textbook designed for the study and communication ofecological principles.(444

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYINVESTIGATING SCHOOL MATHEMATICS, Second Edition. Eicholz, Robert E.; O'Daffer,Phares G.; and Fleenor, Charles R. Menlo Park, California: Addison-WesleyPublishing Company, Inc., 1976. (Addison-Wesley Publishing Company, Inc.,2725 Sand Hill Rd.• , Menlo Park, CA 94025)cloth; textbook series; colorThis is a textbook series for grades 1-6. The books are colorfullyillustrated and include the usual topics in mathematics. A spiralapproach is used for most topics and many thought-provoking extensionsare included.INVESTIGATING VARIATION, Teacher's Edition. Intermediate Science CurriculumStudy. Morristown, New Jersey: Silver Burdett General Learning Corporation,1972. (Silver Burdett General Learing Corporation, 250 James St., Morristown,NJ 07960)93 pp; paper; workbook; color; medium reading levelThis is a workbook with activity-oriented lessons dealing withvariations that occur in man.JUNIOR BIOLOGY: POPULATIONS. Hamilton, Ontario:of Hamilton, n.d. (Board of Education of theW.,Hamilton, Ontario, Canada L8N 3Ll)paper; workbook;b/wBoard of Education of the CityCity of Hamilton, 100 Main St.,Twenty-one activities on the general theme of population aregiven here. Some require biology equipment. Write: 100 Main StreetWest, Hamilton, Ontario, Canada.KEEPING ALIVE, Action Biology Series. weinberg, Stanley L. and Stoltze, HerbertJ. Boston: Allyn and Bacon, Inc., 1974. (Allyn and Bacon, Inc., 470Atlantic Ave., Boston, MA 02210)64 PPi paper; workbook; color; medium reading levelThis workbook gives fifteen activity-oriented lessons dealingwith functions of the body.LABORATORY ACTIVITIES FOR TEACHERS OF SECONDARY MATHEMATICS. Kulm, Gerald. Boston,Massachusetts: Prindle, Weber & Schmidt, In., 1976. (Prindle, Weber, andSchmidt, Inc., 20 Newbury St., Boston, MA 02116)paper;teacher reference; b/wThis is a collection of laboratory activities designed forsecondary students.445

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETY(MODERN SCIENCE EARTH, LIFE, AND MAN. Blane, Sam S.; Fischler, Abraham S.; andGardner, Olcott. New York: Holt, Rinehart and Winston, Publishers, 1971.(Holt, Rinehart and Winston, Publishers, 383 Madison Ave., New York, NY 10017)450 pp; cloth; textbook; color; medium reading levelThis is a textbook that studies the area of science by emphasizinghow man explores the earth, his environment, living organisms and thehuman body.MODERN SPACE SCIENCE. Trinklein, Frederick E. and Huffer, Charles M. New York:Holt, Rinehart and Winston, Publishers, 1961. (Holt, Rinehart and Winston,Publishers, 383 Madison Ave., New York, NY 10017)550 pp; cloth; textbookThis book is a high school text in basic astronomy.MOVE IN ON MATHS 4. Whittaker, Dora. London: Longman Group Limited, 1974.(Longman Group Ltd., 5 Bentinch st., London, England WIM 5RN).63 pp; paper; textbook; color; medium reading levelThis book is one of a series of four student reference textsdesigned to present many varied topics of mathematics."Music and Ma-thematicians since the seventeenth century."MATHEMATICS TEACHER, LXI, No.8 (December, 1968), pp.Brown, J.D.783-787.This is an account of the interest in music of various mathematicians.The article assumes the reader is familiar with suchmathematical topics as differential equations and Fourier series."Music and mathematics." Coxeter, H.S.M. THE MATHEMATICS TEACHER, LXI, No.3(March, 1968), pp. 312-320.This article draws an analogy between music and mathematics,pointing out the need for notation, the use of symmetry, etc. inboth subjects. Several pages are also devoted to ratios of frequenciesof tones and the development of scales from these ratios.THE (MUSIC BY COMPUTERS. Beauchamp, James W. and Foerster, Heinz Von, eds. New York:John Wiley and Sons, Inc., 1969. (John Wiley and Sons, Inc., 605 Third Ave.,New York, NY 10016)clothi teacher referenceThis is a collection of papers on computers in music which werepresented at the 1966 Fall Joint Computer Conference.(450

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYMUSIC, PHYSICS AND ENGINEERING. Olson, Harry F. New York: Dover Publications,Inc.,1967. (Dover Publications, Inc., 180 Varick St., New York, NY 10014)cloth; teacher referenceThis is a handbook for teachers, mUSLCLans, engineers, laymen, andenthusiasts interested in the subjects of speech, music, musical instruments,acoustics, sound reproduction, and hearing."Musical dynamics." Patterson, Blake. SCIENTIFIC AMERICAN, Vol. 231 (November,1974), pp. 78-95.The author here suggests that musicians could and should producegreater variation in loudness of sound to correspond to the levels ofmusical dynamics: pp, p, mp, mf, f, ff. Numerous graphs are displayedin the article."Musimatics or the Nun's Fiddle." Silver, A.L. Leigh. THE AMERICAN MATHEMATICALMONTHLY, Vol. 78, No.4 (April, 1971), pp. 351-357.The article discusses the ratios giving rise to the Pythagoreancomma (see the annotation for the Lewis Salter reference above), theratios in intervals "most pleasing to the ear," and the ratios injust tunings.THE NATURE OF RECREATION. Wurman, Richard Saul; Levy, Alan; and Katz, Joel.Cambridge, Massachusetts: MIT Press, 1972. (Massachusetts Institute ofTechnology Press, 28 Carleton St., Cambridge, MA 02142)76 pp; paper; student reference; b/w; low reading levelThis handbook, which contains many pictures dealing with recreation,is designed to help the reader discover his/her wants andneeds in recreation. Among the pictures is a collection from thenineteenth century.NOISE POLLUTION. Environmental Protection Agency, Office of Public Affairs.Washington, D.C.: U.S. Government Printing Office, 1972. (U.S. GovernmentPrinting Office, Div. of Public Documents, Washington, D.C. 20402)13 pp; paper; student reference; color; high reading levelThe pamphlet covers the adverse effects of noise and some waysto attack noise pollution.THE OFFICIAL ASSOCIATED PRESS SPORTS ALMANAC 1974. Fuller, Keith. New York: DellPublishing Company, Inc., 1974. (Associated Press Newsfeatures, 50 RockefellerPlaza, New York, NY 10020)927 pp; paper; student reference; b/wThe book contains an extensive collection of sports recordsand facts.451

BIBLIOGRAPHYMATHEMATICS AND MUSIC. (Reprint Series) Schaaf, William L., ed. Palo Alto:School Mathematics Study Group, 1967. (Distributed by A.C. Vroman, Inc.,2085 E. Foothill Blvd., Pasadena, CA 91109)25 PP: paper: teacher referenceMATHEMATICSIN SCIENCE AND SOCIETYFour short articles which discuss the commonalitites of Mathematicsand Music: order, harmony, .... and refer to the numerical basis ofmusic.("Mathematics and music." Crow, Warren. SCHOOL SCIENCE AND MATHEMATICS, LXXIV,No.8 (December, 1974), pp. 687-691.This is a short article relating modular arithmetic and transposingmusic. It has good ideas for classroom use.MATHEMATICS AND MY CAREER, "Music and mathematics." Lyon, Howard P. Edited byTurner, Nura Dorothea Rains. Washington, D .. C.: The National Council ofTeachers of Mathematics, 1971, pp. 34-39. (National Council of Teachersof Mathematics, 1906 Association Dr., Reston, VA 22091)collection of essays; teacher referenceSome features common to both mathematics and music are givenhere. Editorial notes include information about the associationof mathematics and music by Pythagoras, Schoenberg and Stravinsky.MATHEMATICS IN WESTERN CULTURE, "The sine of G major." Line, Morris. New York: (",Oxford University Press, 1972, pp. 287-303. (Oxford University Press, 200Madison Ave., New York, NY 10016)paper;teacher referenceA description of the sine curves used to represent musicalsounds is given in this book."Mathematics of musical scales." Malcom, Paul S. THE MATHEMATICS TEACHER, Vol.65 (November, 1972), pp. 611-615.This article discusses the development and mathematical basisof the Pythagorean scale, the just intonation system, the equaltemperamentsystem and some Middle Eastern and Oriental scales.MCDONALD'S ECOLOGY ACTION PACK. Oak Brook, Illinois: McDonald's Corporation, 1974.(McDonald's Corporation, McDonald's Plaza, Oak Brook, IL 60521)29 pp; paper; spirit masterAddress inquiries to: Public Relations Manager, McDonald'sCorporation, One McDonald Plaza, Oak Brook, Illinois 60521.(448

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETY"A merry time with the Moog." Schonberg, Harold C. NEW YORK TIMES (February 16,1969) .A humorous discription of the music in "Switohed-on Bach" (acolumbia record which has music, including the "Brandenburg: ConcertoNo.3, played by the Moog Synthesizer) is given here aswell as the possibilities in the future for electronic music.MICHELSON AND THE SPEED OF LIGHT. (Science Study Series) Jaffe, Bernard. GardenCity, New York: Anchor Books, Doubleday & Company, Inc., 1960. (Doubledayand Company, Inc., 245 Park Ave., New York, NY 10017)195 pp, cloth, teacher reference, b/wThisstudents.importantnon-technical book is easy reading for middle schoolIt gives the life history of A.A. Michelson andearly experiments in light.MIRA MATH FOR ELEMENTARY SCHOOL. (A Mira Math Co. Publication)California: Creative Publications, Inc., 1973. (CreativeInc., P.O. Box 10328, Palo Alto, CA 94303)88 pp, paper, b/w; teacher's guidePalo Alto,Publications,Using a reflective piece of plastic called a "Mira",students areintroduced to properties of reflections, congruence, symmetry and constructiontechniques. The book is designed for grades 4 - 6 and presentsthe activities in a very intuitive matter.MODERN LIFE SCIENCE. Fitzpatrick, FrederickRinehart and Winston, Publishers, 1970.383 Madison Ave., New York, NY 10017)584 pp, cloth, textbook, color, high reading levelL. and Hole, John W. New York: Holt,(Holt, Rinehart and Winston, PUblishers,This is a textbook that approaches the study of living organismsfrom a ecological point of view, emphasizing four content areas--the simple and the complex, the natural community, man and his environment,and production and control.MODERN PHYSICS. Williams, John; Trinklein, Frederick, and Metcalfe, Clark. NewYork: Holt, Rinehart and Winston, PUblishers, 1976. Holt, Rinehart andWinston, Publishers, 383 Madison Ave., New York, NY 10017)724 pp; cloth; textbook; b/wWritten as a high school physics text, this book has manygraphics to illustrate each of the scientific phenomenon discussed.449

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYLABORATORY INVESTIGATIONS IN PHYSICS, Teacher's Edition. Genzer, Irwin, andYoungner, Philip. Morristown, New Jersey: Silver Burdett General LearningCorporation, 1969. (Silver Burdett General Learning Corporation, 250 JamesSt., Morristown, NJ 07960)235 PP: paper: teacher's guide: b/wThis is a high school phyics laboratory manual. In eachexperiment the charts, graphs and tables needed for studentuse are shown very explicitly.(LABORATORY MANUAL FOR THE WORLD OF PHYSICS. Gottlieb, Herbert H. Reading, Massachusetts:Addison-Wesley Publishing Company, Inc., 1973. (Addison-WesleyPublishing Company, Inc., South Street, Reading, MA 01867)232 PP: paper: teacher's guide: b/wThe book is a high school physics laboratory manual in whichall the experiments are designed so they can be solved usingsimple algebra and arithmetic.LIFE SIZE, A MATHEMATICAL APPROACH TO BIOWGY. Gibbons, Rachael F. and Blofield,B. Ann. London: Macmillan Administration LTD, 1971. (Macmillan Administration,(Basingstoke) Ltd., Houndmills, Basingstoke, Hampshire RG2l 2X5)104 PP: paper: teacher's guide: b/wActivities and exercises that show how biology and mathematicscan be interrelated are given here.(THE LIMITS TO GROWTH. Meadows, Donnella, et al. New York: Universe Books. 1972.(Universe Books, 381 Park Ave. S., New York, NY 10016)205 PP: paper: teacher reference: b/wOne of the recent "classics" in describing the predicament ofthe world, this book outlines the consequences of continued growthand the restraints necessary to match population needs and earth'sreSQUrCeS4LISTENER'S GUIDE TO MUSICAL UNDERSTANDING, Fourth Edition. Dallin, Leon. Dubuque,Iowa: wm. C. Bro~ Co., Publishers, 1972. (Wm. C. Brown, Co., Publishers,2460 Kerper BlVd., Dubuque, IA 52001)THE LIVES OF THE GREAT COMPOSERS. Schonberg, Harold C. New York: W.W. Nortonand Company, Inc., 1970. (W.W. Norton and Company, Inc., 500 Fifth st.,New York, NY 10036)600 PP: cloth: student reference: medium-high reading levelThis book is a series of biographies about the great composers.MAKE YOUR OWN MUSICAL INSTRUMENTS. Mandell, Muries and Wood, Robert E.Sterling Publishing Co., Inc., 1957. (Sterling Publishing Company,Park Ave. S., New York, NY 10016)New York:Inc., 419(446

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYTHE MAKING OF MUSICAL INSTRUMENTS. Young, Thomas Campbell. Freeport, New York:Libraries Press, 1969. (Libraries Press, Freeport, NY)MATH IN SPORTS. Des Moines, Iowa: Central Iowa Low-Achiever Mathematics Project.(Central Iowa Low-Achievers Mathematics Project, Board of Education, Des Moines,IA 50310)paper;teacher's guide; b/wThis booklet is designed to take advantage of the naturalinterest students have in sports. Some mathematical topicscovered include computation with whole number, decimals, percents,scale drawings, area and perimeters, and finding averages."Mathematical Games: the arts as combinatoriallike Mozart with dice~" Gardner, Martin.(December, 1974), pp. 132-136.mathematics or how to composeSCIENTIFIC AMERICAN, Vol. 231The use of dice and cards for composing music is discussed alongwith many examples of composers (including Mozart) who used thismethod occasionally."Mathematical-musical relationships: a bibliography." O'Keefe, Vincent. THEMATHEMATICS TEACHER, Vol. 65 (April, 1972), pp. 315-324.This is an extensive 9-page bibliography of articles and booksrelating mathematics and music. Many, but not all, of the referencedworks require a good background in mathematics and music."Mathematicians and music." Archibald, R.C. AMERICAN MATHEMATICAL MONTHLY, Vol.31. (January, 1924), pp. 1-25.A discussion of musical achievements by mathematicians andsome history of scales are given here. Some of this articleis used as a source for the Brown reference listed below.MATHEMATICS ACompanySt., SanHUMAN ENDEAVOR. Jacobs,Publishers, 1970. (W.H.Francisco, CA 94104)529 pp; cloth; textbook; b/w; high reading levelHarold R. San Francisco: W.H.Freeman and Company Publishers,This liberal arts course is an excellent resource hook.Many of the ideas are suitable for or could be adapted formiddle school students.Freeman and660 MarketMATHEMATICS AND LIVING THINGS, Student Text and Teacher Commentary. School MathematicsStudy Group. Palo Alto: The Board of Trustees of the Leland StanfordJunior University, 1965. (Distributed by A.C. Vroman, Inc., 2085 E. FoothillBlvd. , Pasadena, CA 91109)221/170 pp; paper; textbook/teacher's guide; b/wIn this book junior high mathematics is approached throughexperiments with biology. Detailed lessons stressing datagathering, organization and analysis are given.447

BIBLIOGRAPHYOLD OREGON.1975.97403)MATHEMATICS IN SCIENCE AND SOCIETYEugene, Oregon: University of Oregon Alumni Association, Summer,(Old Oregon, Susan Campbell Hall, University of Oregon, Eugene, OR("On being the right size." Haldane, J.B.S. THE WORLD OF MATHEMATICS, Volume 2.Edited by Newman, James R. New York: Simon and Schuster, 1956, pp. 952­957. (Simon and Schuster, Inc., 630 Fifth Ave., New York, NY 10020)688 pp; cloth; teacher reference; b/w"On Being the Right Size" is a delightful five-page essayexplaining relationships on surface area and volume caused bymultiplying the linear dimensions of an organism.ON THE SENSATIONS OF TONE AS A PHYSIOLOGICAL BASIS FOR THE THEORY OFHelmholtz, Hermann. New York: Dover PUblications, Inc., 1954.Publications, Inc., 180 Varick st., New York, NY 10014)MUSIC.(DoverOPTIONS: A STUDY GUIDE TO POPULATION AND THE AMERICAN FUTURE. Horsley, Kathryn,et al. Washington, D.C.: The Population Reference Bureau, Inc., 1973. (ThePopulation Reference Bureau, Inc., 1337 Connecticut Ave. N.W., Washington,D.C. 20036.75 pp; paper; teacher reference; b/w and color(Sierra Elementary Astronomy Series)McGraw-Hill Book Company, 1968. (McGraw-Hill Bookthe Americas, New York, NY 10020)ORBITING THE SUN.50 pp; paper; textbook; medium reading levelThe text, intended for use in fourth, fifth and sixth gradeclassrooms, introduces the structure of the solar system as itis seen from earth_Duke, Robert H. New York:Company, 1221 Avenue of(ORCHESTRAL EXCERPTS FROM THE SYMPHONIC REPERTOIRE FOR TRUMPET, Vol ume I. Compiledby Gabriel Bartold. New York: International Music Company, 1948-1953. (InternationalMusic Company, 511 Fifth Ave., New York, NY 10017)"The original counting systems of Papua and New Guinea."ARITHMETIC TEACHER, Vol. 18, No.2 (February, 1971),Council of Teachers of Mathematics, 1906 AssociationWolfers, Edward P.pp. 77-83. (NationalDr., Reston, VA 22091)THEc452

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYOUTDOOR ACTIVITIES FOR ENVIRONMENTAL STUDIES, Knapp, Clifford, Dansville, NewYork: Instructor Publications, 1971, (Instructor PUblications, 18410 GilmoreSt., Van Nuys, CA 91046)cloth; teacher's guide; b/wThis book outlines several measuring activities--measuring heights,visibility in water, soil compaction, etc.--which can be related toenvironmental concerns aPATTERNS AND PERSPECTIVES IN ENVIRONMENTAL SCIENCE. Washington, D.C.: NationalScience Board, n.d. (National Science Board, 1800 G. St., Washington, D.C.20550)PHYSICAL FITNESS WORKBOOK. Cureton, Thomas Kirk. Urbana, Illinois: Stipes PublishingCompany, 1944. (Stipes Publishing Company, 10-12 Chester St" Champaign,IL 61820)179 pp; cloth; teacher reference; b/wThis is aworkbook for classes in basic physical fitness.PHYSICS, A BASIC SCIENCE, Fifth Edition. Verwiebe, Frank; Van Hooft, Gordan; andSaxon, Bryant, New York: American Book Company, 1970. (American Book Company,450 W, 33rd St., New York, NY 10001)544 pp; cloth; textbook; b/wThis book is written as ahigh school physics text.PHYSICS, A LABORATORY COURSE, Teacher's Edition.and Saxon, Bryant. New York: American Bookpany, 450 w. 33rd St., New York, NY 10001)118 pp; paper; teacher's guide; b/wThis is a high school physics laboratory manual.Verwiebe, Frank; Van Hooft, Gordon;Company, 1970. (American Book Com-PHYSICS, A MODERN APPROACH. Elliot, Paul and Wilcox, William. New York: MacmillanPublishing Company, 1959. (Macmillan Publishing Company, Inc., 866 Third Ave.,New York, NY 10022)686 pp; cloth; textbook; b/wThis book is ahigh school physics text.PHYSICS DEMONSTRATION EXPERIMENTS, Vol umes I and II. Meiners, Harry F. New York:The Ronald Press Company, 1970. (The Ronald Press Company, 79 Madison Ave.,New York, NY 10016)1395 pp; cloth; teacher reference; b/wThis is a large collection of demonstration and experimentscovering the areas of mechanics and wave motion, heat electricityand magnetism, optics, and atomic and nuclear physics.453

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYTHE UNIVERSE IN MOTION, Book 2, Guidebook. (The University of Illinois AstronomyProgram Series) Elementary-School Science Project. Co-directed by Atkin,Myron J. and Wyatt, Stanley P. Jr. New York: Harper and Row Publishers, 1969.(Harper and Row Publishers, 10 E. 53rd St., New York, NY 10022)96 pp; teacher's guideThe University of Illinois Program is a series of six booksthat introduces astronomy to upper elementary and junior highstudents. The major concepts in each book are explored throughnumerous student activities as well as through the developmentof models that explain astronomical phenomena. The series isan excellent resource for applications of mathematics to astronomy.WELL-BEING, PROBING THE NATURAL WORLD, Level III, Teacher's Edition. IntermediateScience Curriculum Study. Morristown, New Jersey: Silver Burdett GeneralLearning Corporation, 1972. (Silver Burdett General Learning Corporation,250 James st., Morristown, NJ 07960)143 pp; paper; workbook; color; medium reading levelThis is a workbook with activity-oriented lessons dealingwith the senses and functions of the body, and how these areaffected by food, smoking, alcohol, and other drugs."We made it and it works! the classroom construction of sundials. Wahl, M.Stoessel. THE ARITHMETIC TEACHER, Vol. 17, No. 4 (April, 1970), pp. 301­304. (National Council of Teachers of Mathematics, 1906 Association Dr.,Reston, VA 22091)magazine; teacher referenceThis is a publication by the National Council of Teachers ofMathematics which contains many suggestions for classroom activitiesfor elementary and junior high school students.(("What is electronic music?" Seay, Albert. MUSIC JOURNAL, Vol. 21 (March, 1963).pp. 26, 60-6l.The article gives a brief overview of the development ofelectronic music from 1906 to 1963.WHY YOU'RE YOU, Teacher's Edition. Intermediate Science Curriculum Study. Morristown,New Jersey: Silver Burdett General Learning Corporation, 1972. (SilverBurdett General Learning Corporation, 250 James St., Morristown, NJ 07960)134 pp; paper; workbook; color; medium reading levelThis is a workbook with activity-oriented lessons on thestudy of heredity.(458

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYWORDS IN SPACE • .. DIMENSIONS ON A VAST SCALE, Teacher Folio-Investigation 9,(Time, Space, and Matter .•. Investigating the Physical World Series)Secondary School Science Project. Princeton, New Jersey: Webster Division,McGraw-Hill Book Company, 1969. (McGraw-Hill Book Company, 1221 Avenue ofthe Americas, New York, NY 10020)56 pp, paper; teacher's guide; b/wThe booklet contains background information and suggests activitiesfor investigating the size, distance and motion of objects in the solarsystem.THE WORLD ALMANACInc., 1974.NY 10017)& BOOK OF FACTS 1975. New York: Newspaper Enterprise Associates,(Newspaper Enterprise Associates, Inc., 230 Park Avenue, New York,976 pp; paper; teacher reference; colorThis book contains a collection of world facts and events.published annually.It isTHE WORLD OF PHYSICS. Hulsizer, Robert and Lazarus, David. Reading, Massachusetts:Addison-Wesley Publishing Company, 1972. (Addison-Wesley Publishing Company,Inc., South Street, Reading, MA 01867)518 pp, cloth; textbook; b/wThis book, written as a high school physics text, is full ofhand-drawn graphics which are easy to understand.YOUTH PHYSICAL FITNESS: SUGGESTED ELEMENTS OF A SCHOOL-CENTERED PROGRAM. Washington,D.C.: United States Government Printing Office, 1967. (U.S. GovernmentPrinting Office, Div. of Public Documents, Washington, D.C. 20402)109 pp; paper; teacher reference; b/wThis book has exercises to evaluate and promote physical fitnessof students.459

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETY"The role of music in Galileo's experiments." Drake, Stillman. SCIENTIFICAMERICAN, Vol. 232, No.6 (June, 1975), pp. 98-104.An explanation is given on how Galileo could have used a simplemarching song to mark equal lengths of time (since no device couldmeasure less than a second) and rubber bands on an inclined plane·(much like the adjustable frets on the viola da gamba's of his day)to measure distance traveled.(SCHOOL MATHEMATICS I. Eicholz, Robert E.; O' Daffer, Phares G; Brumfiel, CharlesF.; Shanks, Merril Eo; and Fleenor, Charles R. Menlo Park, California:Addison-Wesley Publishing Company, Inc., 1971. (Addison-Wesley PublishingCompany, Inc., 2725 Sand Hill Rd., Menlo Park, CA 94025)cloth; textbook series; b/wimediurn reading levelSCHOOL MATHEMATICS I is the 7th grade text that complements theELEMENTARY SCHOOL MATHEMATICS series from the same publishers.SCIENCE & MUSIC. Jeans, Sir James. New York: Dover Publications, Inc., 1968.(Dover Publications, Inc., 180 varick St., New York, NY 10014)paper;teacher referenceFirst published in 1937, this book is written for the amateuras well as the serious student of music to explain precisely andin a non-technical way the physical basis of musical sounds.SCIENTIFIC AMERICAN. New York: Scientific American, Inc. (WID. H. Freeman andCompany Publishers, 660 Market st., San Francisco, CA 94104)journal; teacher reference; b/w and colorThe September, 1970 issue is devoted to the biosphere; theSeptember, 1971 issue to energy.(SEASONAL STAR CHARTS, A COMPLETE GUIDE TO THE STARS.Press, a division of Hubbard Scientific Company,Box 442, Northbrook, IL 6005223 pp; paper; stUdent reference; high reading levelNorthbrook, Illinois: Hubbard1972. (Hubbard Press, P.O.The book includes a polar star map that shows the basic starsand constellations visible in the northern hemisphere with the unaidedeye as well as eight seasonal star charts and information about themajor constellations."Some relationships between music and mathematics." Cohen, Joel E. MUSIC EDUCA­TORS JOURNAL, Vol. 48 (September, 1961), pp. 104-109.A discussion of the kinship of mathematics and music is givenin this article. Included, also, is the use of abstract notation,the creative process, and the search for beauty, which are commonto both disciplines.456

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYSOURCEBOOK FOR ENVIRONMENTAL EDUCATION. Vivian, V. Eugene. St. Louis, Missouri:C.V. Mosby, 1973. (C.V. Mosby Company, 11836 Westline Industrial Dr., St.Louis, MO 63141)teacher referenceSRA MATH APPLICATIONS KIT. Friebel, Allen C. and Gingrich, Carolyn Kay. PaloAlto, California: Science Research Associates, Inc~, 1971. (Science ResearchAssociates, 259 E. Erie St., Chicago, IL 60611)280 activity cards; student reference; b/w; low reading levelThe MATH APPLICATIONS KIT is a collection of activity cardsarranged into the following areas: appeteasers, science, sportsand games, occupations, everyday things, and social sciences.The cards are open-ended investigations, and good referencematerial and reference cards are also included in the kit."Teacher's Guide to the Energy Crisis." Sartwell, Joyce G. and Abell, Richard P ..TODAY'S EDUCATION, January-February 1975, pp. 90-94. (National Education Asso~ciation, 1201 16th St., N.W., Washington, D.C. 20036)TEACHING POPULATION CONCEPTS. King, Pat and Landahl, John. Olympia, Washington:Office of Environmental Education, Superintendent of Public Instruction, 1973.(Office of Environmental Education, Washington State Dept. of Public Instruction,Old Capitol Bldg., Olympia, WA 98504)60 pp; teacher referenceThis booklet is oriented more toward high school. It is nowavailable from John Landahl, 1207 N.E. 103rd St., Seattle, Washington98125.TRACK AND FIELD ATHLETICS. Bresnahass, George T.; Tuttle, W.W.; and Cretzmeyer,Francis x. St. Louis: C.V. Mosby Company, 1964. (C.V. Mosby Company, 11836Westline Industrial Drive, St. Louis, MO 63141)424 pp; cloth; teacher reference; b/wThis book describes techniques for athletes in track and field.THE UNITED STATES STATISTICAL ABSTRACT 1975.Printing Office, 1975. (U.S. GovernmentDocuments, Washington, D.C. 20402)1050 pp; cloth; teacher reference; b/wWashington, D.C.: U.S. GovernmentPrinting Office, Div. of PublicThe book is a united States Government yearly publication ofstatistics on social, political and economic conditions of theUni ted States•457

BIBLIOGRAPHYMATHEMATICS IN SCIENCE AND SOCIETYTHE PHYSICS OF MUSICAL SOUND. Josephs, Jess J. Princeton, New Jersey:Nostrand Company, Inc., 1967. (Van Nostrand Reinhold Company, 450St., New York, NY 10001)paperj teacher referenceThis monograph (published for the Commission of College Physics)explains what the science of acoustics has been able to contribute tothe understanding of music.D. VanW. 33rd.(THE PHYSICS TEACHER. American Association of Physics Teachers, State Universityof New York at Stoney Brook. New York: American Institute of Physics.magazineTHE PHYSICS TEACHER, a publication of the American Associationof Physics Teachers, is published monthly through the academicyear. Every issue has a variety of articles which describe experimentsand teaching techniques for physics.PHYSIOLOGICAL ADAPTATION. Biological Sciences Curriculum Study. Segal, Earl, andGross, Warren J. Lexington, Massachusetts: D.C. Heath and Company, 1967.(D.C. Heath and Company, 125 Spring St., Lexington, MA 02173)109 pp; paper; workbook; b/w; high reading levelThis is a workbook designed to allow students to discover variousideas dealing with the physiological adaptations performed by the body."pi in the key of C." Rhoades, Patrick Alan. INDIANA MATH NEWSLETTER (October,1966), pp. 3-4.(POLLUTION' PROBLEMS, PROJECTS AND MATHEMATICS EXERCISES, 6-9 (Bulletin NO. 1082).Madison, Wisconsin: Wisconsin Department of Public Instruction, n.d. (WisconsinState Dept. of Public Instruction, 126 Langdon St., Madison, WI 53702)84 pp; paperThis is a teacher-written collection of problems and projects.Exercises are grouped under whole numbers, rational numbers, realnumbers, percent and proportion, measurement and statistical measuresand grapps.PRINCIPLES OF ENVIRONMENTAL SCIENCE. Watt, Kenneth. New York: McGraw-Hill BookCompany, 1973. (McGraw-Hill Book Company, 1221 Avenue of the Americas, NewYork, NY 10020cloth; teacher reference; b/w(454

BIBLIOGRAPHY MATHEMATICS IN SCIENCE AND SOCIETYPROBING THE NATURAL WORLD, Volume I. Intermediate Science Curriculum Study.Morristown, New Jersey: Silver Burdett General Learning Corporation, 1970.(Silver Burdett General Learning Corporation, 250 James St., Morristown, NJ07960)552 pp; cloth; textbook; b/wPROBING THE NATURAL WORLD is one of two books in a series whichcovers the normaL junior high science material, but with an extensiveuse of hands-on laboratory work using homemade equipment.PROJECT JONAH. Box 476, Bolinas, California 94924.PROJECT R-3. Hodges, E.L., ed. San Jose, California: T.M.T.T., 1974.Hodges, 990 Asbury, San Jose, CA 95126)looseleaf; worksheets; b/wFour packets of student pages and a packet of answer sheetsand suggested forms for grading, record keeping, etc. make up thematerials. The four packets cover whole numbers, fractions, decimalsand percents.(E .L.PSSC PHYSICS, Third Edition. Haber-Schiam, Uri; Cross, Judson; Dodge, John; andWalter, James. Lexington, Massachusetts: D.C. Heath and Company, 1971.(D.C. Heath and Company, 125 Spring St., Lexington, MA 02173)674 pp; textbook; cloth; b/wwritten as a high school physics text, this book has manynice graphics, including some interesting stop-action photos."A rage over an extra cent." Salter, Lewis S. AMERICAN MUSIC TEACHER, Vol. 3(March-April, 1954), pp. 1,18-19.This article presents a somewhat humerous discussion of aninaccurate number often cited in music. The author explains (bycomputing in three different ways) that the Pythagorean comma,the interval by which the twelfth fifth above a given tone failsto coincide with the seventh octave above the same tone, measurescloser to 23 cents (a unit for measuring intervals) than to 24cents as generally cited. Extensive use of logarithms is used inthe article. It is helpful if the reader is quite familiar withthe construction of scales.RATIO & PROPORTION REVISITED. Shulte, Albert, et al. Pontiac, Michigan: OaklandSchools, 1970. (Oakland County Mathematics Project, Oakland Schools, 2100Pontiac Lake Rd., Pontiac, MI 48054)paper; teacher's guide; b/wOne of a series which is very student oriented and which includesmany activitiesa455

BIBLIOGRAPHYFILMSMATHEMATICS IN SCIENCE AND SOCIETY(IlArt from Computers." NBC Educational Enterprises, Canada, 1971 .. 10 min."Dance Squared." National Film Board, Canada, IFB, 1963. 4 min."Discovering Electronic Music .. " Bailey Film Association, 1970. 23 min."Donald in Mathemagic Land." Disney Productions, 1959. 20 min."New Sounds in Music." Churchill Films, 1969. 22 min."Notes on a Triangle." National Film Board, Canada, IFB, 1967. 5 min.lIPretty Lady and the Electronic Musicians." Xerox, 1972. 14 min.IlTrio for 3 Angles." National ,Film Board, Canada, IFB, 4 min.RECORDSCOMPUTER MUSIC from the University of Illinois, Hiller/Isaacson/Baker, Helioda(H/HS 25053). Metro-Goldyn-Mayer, 1350 Avenue of the Americas, New York, NY10019Side 1: Illiac Suite for String QuartetSide 2: Computer Cantara(COMPUTER MUSIC from the Computer Center of Columbia and Princeton Universities.Randall/Vercoe/ Dodge, Nonesuch Recerds· (H-71245), 15 Columbia Circle, NewYork, NY 10023.MUSIC FROM MATHEMATICS played by the IBM 7090 Computer and Digital to Sound Transducer,Decca Records (DL 9103).AUDIO-TAPES"Science and the Drama of Music Series,1I part I, STAO, 1954, 18 7-inch reels, 7.5ips. 30 min. each.(460

Page Number78 (1) noon; sunrise or sunset (2) 10:38 a.m. (3) 2:52 p.m.78 (la) weaker(2a) weaker(3a) weaker(4a) weaker(5a) weaker(lb) 13.44 m (44' 1 1/2")(2b) 14.30 m (46' 10 9/10")(3b) 2.56 m (8' 4 6/7")(4b) 4.58 m (15' 5/12")(5b) .93 m (3' 37/48")79 First Quarter ~ Waxing Crescent ~New • Waning Crescent • Third QuarterctWaning Gibbous0 Full0 Waxing Gibbous 081(1) 21 km; 20 km(2) 33 m (3) 59 0(4) 1560 m83(6) 1900 km; 1700 kmmately 400,000 km(1) 20,000 km(2) The distance is approxi-84 (le) yes; noon; Yes, the angle increases as the shadow increases(2b) 100 m85 (3) 39500 km (4) 12700 km (5) 12600 kmHint: Divide the circumference by pi.88 (2) Lengthening the base line increases the greatest distance that can bemeasured.95 (2c) 3.7 cm; .7 a.u. (3) 2 cm; .4 a.u.96 (ld) 7.5 em9799100101(1) 375,000 km (2) 42,000,000 Oa) .28 (3b) 1 a.u.;150,000,000 km (3c) Mercury - 59,000,000 km; Venu~ - 108,000,000 km;Earth - 150,000,000 km; Mars - 228,000,000 km; Jupiter - 780,000,000 km;Saturn - 1,431,000,000 km; Uranus - 2,877,000,000 km; Neptune ­4,509,000,000 km; Pluto - 5,928,000,000 km(2f) Angular size decreases as distance increases.(2) The diameter of the moon is about 3600 km.(4) The diameter of the sun is about 1,440,000 km.461

Page Number3 It would be at its original place. (299translationdown 1 placereflectionof previousmeasurerotationreflectionrotation33003013074 Answers vary.(II) The half-full glass gives a lower tone. (12) Note: It might bedifficult to get the entire scale. This will depend on the shape andsize of the glasses. (13) varies (113) Students should see amarking pattern like below. It is easy to see where the half-steps are.~:::.:.;.:;;'J.·.:.i:.: ..... ,.,.,~~n=·@(I) anSwers will vary (II) If the music is written in the key of C,probably C will be used frequently. E, F, and G might also occur frequentlyin music in the key of C because they are perfect thirds, fourths,and fifths. Any other note in the major scale might also be used. Ifthe music is written in a minor key, or is more complicated in otherways, it will be difficult for most students to determine why a particularletter occurs more frequently. (III) C(Ia) AB, BA, BB (Ib) AA, AB, AC, BB, BA, BC, CC, CA, CB; 9 or 3 x 3(Ic) 16 or 4 x 4; 25 or 5 x 5; 36 or 6 x 6; 49 or 7 x 7; 144 or 12 x 12(IIa) AAA, AAB, ABA, BBA, BAA, BAB, ABB, BBB; 8 or 2 x 2 x 2 (lIb) 3 3or 3 x 3 x 3; 4 3 or 4 x 4 x 4; 53 or 5 x 5 x 5; 6 3 or 6 x 6 x 6; 7 3 or7 x 7 x 7; 12 3 or 12 x 12 x 12 (III) 7 5 or 7 x 7 x 7 x 7 x 7((466

Page Number310311313314(VII) B is A read backwards so B is the retrograde of A. C is an inversionof A about G or it can be viewed as a transposition and then aninversion about the first note. D is the retrograde of C (A could betransposed, inverted and read' backwards to obtain C).(I) 4 (IIa) higher (lIb) lower (IVb) lower pitch; softer(IVc) lower pitch; louder (IVd) same pitch; softer (IVe) higherpitch; same (VIla) violin; lower pitch; louder (Vllb) flute; samepitch; louder (Vllc) clarinet; same pitch; louder (Vlld) tuningfork; higher pitch; softer(1) harp; about 33; about 3300 (2) timpani - 73 to 194 or tubularbells - 261 to 698 (Both range over 18 half-tones even though the frequencyrange of the bells is greater than the timpani.) (3) varies(4) french horn; bass (5) soprano(1) 32.703; 65.406; 130.81; 261.63; 523.25; 1046.5; 2093.0; 4186. Thenumbers double with each octave higher in pitch. There is some "roundoff" error. (2) 48.999; 97.999; 196.00; 392.00; 793.99; 1568.0;3136.0; yes (3) yes(4) C65.406130.81261. 63523.251046.5E82.407164.81329.63659.261318.5G97.999196.00392.00783.991568.0E fC1. 261.261.261.261.26G T C1.51.51.51.51.5F43.65487.307174.61349.23698.46A55.00110.00220.00440.00880.00C65.406130.81261. 63523.251046.5A fF1.261.261. 261.261.26C fF1.51.51.51.51.5315319(1) possibilities: fire alarm, gun shot, explosion (2) possibilities:bird twittering, soft breeze, baby breathing, whisper (3) varies(4) An ordering is given On the second page. Students might questionthe placement of the food blender or other sounds. Certainly the soundlevel of many of the items could vary greatly. (5) 50, 60, 70••• 200(6) 10 3 ; threshold of pain: 10 12 ; lethal level 10 18 ; threshold of hearing:100. The powers of the column will range from 100, 10 1 ••• 10 2 °.(7~ whisper; conversation; subway (8) If the relative intensity is10 , the decibel reading is N x 10. (9) rock group (lOa) 3000;guesses will vary (lOb) No, its an advertisement and they are tryingto impress people (lOc) answers vary(1) fff: very very loud; ppp: very very soft (2) 20; 42; no(3) soft recording: 35; .loud recording: 85; line symphony: 105 (Ofcourse readings could vary greatly.) (4) meaning "greater than~II467

Page Number245 (1) .e1ectric train; car with 1 person (2) planes are very fast(3) convenience; the cost of driving oneself in the U.S. is still reaSonable(4) this is average number of people observed per car(5) column 4 = column 2 x column 3 (6) 70 (7) 36; no (8) lessair pollution; safer, less crowded street and highways(246 (1) 10 kJ.n1R­(5) 1100 kJ.n247 (1) 10; 8.6(6) 13.7(2) 35 (3) 6 (4) they help control air pollution(2) about 13 minutes (3) about 22 minutes (5) 39.6264 answers will vary according to performance266 (I) The counts will probably vary, but if students count accurately, theaverage should be about 33 1/3. (II) about 45 (III) about 78(IV) The marker on the label and the marker on the edge of the recordboth make 331/3 revolutions per minute. The inside marker travelsaround a smaller circle so it does not travel as far, but its angularvelocity is the same; that is, both markers sweep out the same angle inany given amount of time. (a) 11 2/3 rpm (b) 11 2/3 rpm (c) answersvary (d) 157.5 (e) 700 (f) If an outer band has doublethe diameter of an inner band, the needle goes twice as far. The recordingprocess takes care of the difference.268(1) 23,571 pounds persquare inch{0002 in. x .0007 in. = .00000014x 1bs. 1 in. 2.0033 lbs. = .00000014 in. 2in. 2 }(2) xl. 2 or about 69 pounds per in. 210,000 144 In.(3) the phonograph needle (4) the needle wears out the record. Itvery light, but all of its weight is exerted on a very fine point.seems(269 Answers will vary depending on the music chosen.270 Crossword puzzle: Across: (2) duet, (4) quartet, (6) triplet, (7) trio(9) triad, (11) octave, (12) monotone Down: (1) sextet, (3) quintet,(5) hexad, (8) octet (10) solo Matching puzzle: e, d, h, g, b, f, a, c285287 incorrect measures are: 1, 3, 5, 6, 8, 12, 13, 14, 17, 18, 19, 23, 24288 (Ib) two beats(If) seven beats(c) five beats(g) nine beats(Id) three beats(Ih) eight beats(Ie) six beats(464

Page Number289 square, trouble, tied, square, step, severe, base, bar, greater than,flat, right, triple, hold, rest, cut, equal, try--angle, sharp, accent,staff, base, natural, note, less than, minus, rest, divide, base. lastline: see sharp, be flat-,_••j~,.~.~.·~~~~I~·~.~.~.~.=t= . • -.- •••• .... ...21 .......... ] ........ E2- ..... 1 ........•3~t~~~297I t¥ x... I • )( • t?tx• I ....... X ... •• I~.~~"'~",~~x~"'~~.~-~~~465

Page NumberIII (1) Yes «2) Using the graph, the distance is 2.1 a.u.Law we see that 2.1 cubed is 9.261, which rounds to 9.5.2; 4.6; 11.2 (4) 1000 yearsUsing Kepler's ((3) 2.8; 2.5;113 (1) The greater a planet's distance from the sun,speed. (2) The square of the numbers in columnof the numbers in column 1. (3) 7.5 km/sec.(5) 8 Years (6) No orbital speed.the slower its orbital2 are the reciprocals(4) 1/4 a.u.181190Word Hunt: Math: seven, even, percent, divide, sine, line, pi, sum,terms, circle, square, union, set, tangent, ration, factor, fraction,minus, map, ray, cone, ten, cube. Ecology: recycling, rats, ecology,ash, tin, depletion, oilslick, land, landfill, man, detergent, soap,litter, ghetto, air, river, pollution, mud, rot, life, car, resource,sun, conservation, slum, junk, rock.(2) 60; 3600; 86400 (3) 31,536,000 (4) almost 127 years(5) 197,200; 71,978,000; no (6) 1,584,900; yes196 (2) 9; 81 (3) 161915; about 1880masS starvation.(5) Only in the first generation. (6) 1968;(7) Number of children people have, wars, epidemics,199201(1) 1000 (3) California; 9, 9 (4) New York; Texas(6) 5; lists will vary (7) 9, lists will vary(5) about 1/4(1) 9 kg (2) 1/2 x height before; 1/8 x mass before (3) 5120 kcal(based on mass) or 2560 kcal (based on surface area) (4) 20 weeks(based on mass) or 10 weeks (based on surface area) (5) 250 kg(6) 1/32; new:old = 1:32768 (7) 128 (based on mass or volume) or32 (based on height)(208 (2) In 1976 - 68,800 km, more than 1 1/2earth. (3) 64,500,000,000; 64,500,000.high death rate. (5) 21,500,000times the distance around the(4) 154; 16,555,000,000;209(1) They feed mainly on fish, which have a fairly high concentration ofDDT. For many poisons, animals higher in a food chain contain greaterconcentrations. (2) .0126 (3) From eating fish from DDT infestedareas: .0063 (4)Fish are the osprey's primary food. (5) 1,120,000;Primarily because of a bDT campaign against mosquitos in countries wheremalaria was common.213 (5) 26.7; 34.4; 41.1; 46.2; 51.0 (6) The number of families214 (4) The increase is only slight (5) The heaviest smokers; 65%(6) People who have never smoked (7) In some cities, one breathes theequivalent of 38 cigarettes per day.(462

Page Number216 (1) .84 (2) 153 million (3) 13,975,000,000California have severe air pollution problems.for each kind of pollutant.(4) Some areas of(5) They were higher223 (3) 7,525,000,000 {full loads of clothing(5) 43,000,000,000, {;(4) Not letting water run needlessly,only, taking quick showers...86,000,000,000 {washing224 (2) 31200 {(7) 3600 {(3) 40000 { (4) 4000 {lear (5) 80 { (6) 200 {(8) Large amounts of water are required to make steel.226 (1) 48 (2) Water from rainfall and streams which flow into the oceansand seas replace the evaporated water.227 (1) Gulf (2) Northeast; Gulf; Missouri; Columbia (3) Southeast;Columbia (4) Ohio (83%); Great Lakes (41%); Southeast (38%); California(30%); Columbia (19%); Gulf (17.95%); Northeast (17.92%), MiddleAtlantic (17.6%); Missouri (17%)228 (4a) 37,900,000229 (1) alewife, whitefish231 (4) 4.3 (5) $781.42; about $3330 (6) 685,000 ha; $535,272,700(7) 310,000,000; much of it is exported232 (1) true (2) true (3) false (4) true234 (la) 912.5 kg (6) open dumping; It is the cheapest and easiest method.235 (1) increased (2) true (3) about 33% (4) .44¢No, we don't know how many cans were made in 1970.(5) 3.3;238 (1) 5000 (2) 90,000 ha (3) 2,880,000 ton (4) 1700 times asmuch per hectare (5) strip farming, contour plowing, keeping plantcover on the land. Students can stay on trails in parks, fill in erosioncausedditches, avoid wearing out the grass in play areas...241 (1) 93 (2) 94% in 1970 to 71% in 2000 (3) no. Energy use is expectedto increase so much that the lower percentage will still mean moreenergy. (4) 286 (6) fossil fuels243 (1) 549; 269 (5) 195,000,000,000,000463

Page Number388 (a) pentagons and hexagons; 12 pentagons, 20 hexagons; 32 (b) 2; 1;All vertices are surrounded by one pentagon and two hexagons; 50; Novertices of a pentagon are shared by another pentagon, so there are5 x 12 = 60 vertices (c) 90 (d) no400 (5) 57; 167.5; tall, average (6) 29; 132.5; short, light409 (6) Men's by 2.2 pounds (1 kilogram)events.(7) Women don't compete in these410416(3a) about 4 minutes 28 secondsNew York(3b) about 37 seconds(468