BRITISH COLUMBIA ASSOCIATION OF MATHEMATICS TEACHERS … · BRITISH COLUMBIA ASSOCIATION OF...

BRITISH COLUMBIA ASSOCIATION OF MATHEMATICS TEACHERS

N EWSLETTE R/JOU RNAL

VOLUME 16, NUMBER 4 JUNE 1975

BCAMT EXECUTIVE 1974-75

PAST PRESIDENT PRESIDENT J. Michael Baker Alan Taylor 11225 - 87th Avenue 7063 Jubilee Street Delta, B.C. V4C 2Z6 Burnaby, B.C. V5J 4134 594-8127 (home) 434-6315 (home) 588-1258 (school) 936-7205 (school)

PRESIDENT-ELECT VICE-PRESIDENT John Epp Roger Sandford 1612 Wilmot Place R.R. 1, Tzouhalem Road Victoria, B.C. V813 5S4 Duncan, B.C. V9 1M3 592-2388 (home) 746-6418 (home) 478-5548 (school) 746-4435 (school)

SECRETARY SECRETARY-ELECT Mrs. Florine Carlson Ian C. DeGroot 3043 Spuraway Avenue 3852 Calder Avenue Coquitlam, B.C. V3C 2E4 North Vancouver, B.C. V7N 3S3 941-3790 (home) 980-6877 (home) 941-3408 (school) 987-7178 (school)

TREASURER ELEMENTARY REPRESENTATIVE Bill Dale Mrs. Grace Dilley 1150 - 17th Street 2210 Dauphin Place Courtenay, B.C. V9N 1Z7 Burnaby, B.C. V5B 4G9 338-5159 (home) 299-9680 (home) 334-2428 (school) 594-6433 (school)

NCTM REPRESENTATIVE IN-SERVICE SPECIALIST Tom Howitz Dennis Hamaguchi Faculty of Education, UBC 3807 - 22nd Avenue 2075 Wesbrook Place Vernon,B.C. Vancouver, B.C. V6T 1W5 V1T 1H7 325-0692 (home) 542-8698 (home) 228-5203 (UBC) 542-3361 (school)

PUBLICATIONS CHAIRMAN PUBLICATIONS CHAIRPERSON-ELECT Bill Kokoskin Shirley McGill 1341 Appin Road Box 458 North Vancouver, B.C. V7J 2T4 Clinton, B.C. VOK 1 KO 988-2653 (home) 456-7584 (home) 988-3161 (school)

PRIMARY REPRESENTATIVE SUMMER WORKSHOP 1975 Marion Harvey Doug Owens 946 Gatensbury Street Faculty of Education, UBC Coquitlam, B.C. 2075 Wesbrook Place, V3J 5J3 Vancouver, B.C. V6T 1W5 939-5488 (home) 596-0718 (home) 939-4201 (school) 228-4808 (UBC)

NORTHWEST NCTM CONFERENCE CO-ORGANIZERS Trevor Calkins John Epp 1623 Amphion Street 1612 Wilmot Place Victoria, B.C. V813 4Z5 Victoria, B.C. V813 5S4 592-4463 (home) 592-2388 (home) 592-1205 (school) 478-5548 (school)

Inside This Issue....

5 President's Final Message....................Alan Taylor

8 Your Executive Committee...................Bill Kokoskin

9 List of Publishers' Representatives........Bill Kokoskin

13 A Summer Institute in Computer Science.................. ...................................David F. Robitaille

15 Treasurer's Report 1974-1975 ...................Bill Dale

17 A Book Worth Reading: 'Why Johnny Can't Add'...........

.............................................Barry Gear 20 Problems for Computer Science..................Frank Ebos

23 Fourth Mathematics Summer Workshop......................

24 In-service Education: Views of Teachers............NCTM

28 Highlights from Across the Country.............Ed Murrin

30 Broad Spectrum Mathematics Project...................... .........................Peter Weygang, Alan C. Madgett

36 Preliminary Drafts of Proposed Courses for Years 11 and 12 ...............................Math Revision Committee

47 Distributor Cap Mathematics.............William J. Bruce

51 Games People Play (Elementary) ................Frank Ebos

60 Games People Play (Secondary) .................Frank Ebos.

66 Experiences with Functions...................J.B. Harkin

77 New Books Across My Desk...................Bill Kokos kin

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President's Final Message

The major goal of the BCAMT for 1974-75 was to promote, ini-tiate and develop in-service activities in mathematics edu-cation.

To realize our goal we had to attain certain objectives. First, needs were established; next, communication facilitat-ed and finally, action was initiated and followed through.

Establishment of Needs

The BCAMT, in conjunction with UBC's Faculty of Education and the BCTF, undertook a program to establish in-service needs of B.C.'s math teachers. A questionnaire was developedand sent to each school. A pattern of needs emerged at each school level, and a geographical breakdown of the results determined regional needs. Next, our in-service specialist, Dennis Hamaguchi, contacted local in-service personnel to point out needs in their areas and to offer assistance in the organization of workshops.

Facilitating Communication

Our newsletter/journal, Vector, remained the most effective avenue of communication. The editor selected content to provide relevant and informative reading for teachers at all levels. Due to significant cost increases, the number of issues was limited to four.

To provide balance in the executive and to gain input from primary and intermediate levels, we appointed Marion Harvey primary representative and Grace Dilley intermediate repre-sentative.

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A new avenue of communication with the Mathematics Revision Committee was developed through the appointment of our editor, Bill Kokoskin, to that body. Direct input and feedback is facilitated through this liaison.

We maintained a liaison with teachers of mathematics at the post-secondary level. In addition to direct contact with UBC through Dr. Tom Howitz, we participate in meetings sponsored by the British Columbia Committee on the Undergraduate Pro-gram in Mathematics (BCCUPM).

In-service Activities

In August 1974, we sponsored a two-day workshop at Queen Elizabeth Secondary School in Surrey. An excellent program was organized by the Summer Workshop Committee under the chairmanship of Dr. Walter Szetela. The response was over-whelming. More than 400 teachers participated.

Plans are now under way to offer a summer workshop in August 1975 at New Westminster Secondary School. Dr. Doug Owens is chairman of this year's committee. The main theme revolves around curriculum revision.

Trev Calkins and John Epp are busily organizing the 15th Northwest Mathematics Conference. It will be held in October 1976 at Victoria. More than 1,200 registrants from B.C., Washington and Oregon are expected.

We encourage PSA chapters to form and plan in-service activities in their districts. Help is offered in the form of advice, in-service personnel and money. At present, locals are active in Coquitlam, Victoria and Nelson.

Conferences and Workshops:

1. Summer Mathematics Workshop - held August 27 and 28 at Queen Elizabeth Secondary school in Surrey. More than 400 participated.

2. 15th Northwest Conference - to be held in October 1976 at Victoria. Planning is under way with more than 1,200 parti-cipants expected.

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3. Summer Mathematics Workshop - to be held in August 1975. The planning committee has been formed, and the program is under way.

4. Thinktank - to be held in spring of 1975. Members of the executive will participate. The object is to determine the direction of mathematics education in the province and the implications of decentralization.

Meetings:

1. September 14, 1974 - Executive Meeting at the B.C. Teachers' Building.

2. November 2, 1974 - Executive Meeting at the B.C. Teachers' Building.

3. December 7, 1974 - Executive Meeting at the B.C. Teachers' Building.

4. February 1, 1975 - Executive Meeting at the B.C. Teachers' Building.

5. March 22, 1975 - Annual General Meeting and Executive held at the B.C. Teachers' Building.

6. June 7, 1975 - Executive Meeting at the B.C. Teachers' Building.

7. Thinktank Session - time and location to be determined.

Respectfully submitted,

Alan R. Taylor, President

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Your Executive-Committee

BILL KOKOSKIN

Bill is the Treasurer of the BCAMT. He obtained his B.Ed. (Soc.) at UBC

in '64. His teaching experience spans three districts (North Vancouver, Coquitlam and Courtenay), six Grade levels '(8 to 13 inclusive) and two subject areas (Mathematics and Biology). For the past five years, Bill has taught math at the Courtenay Junior Secondary School. This September, he will be at the George Vanier Senior Secondary School.

BILL DALE

Last summer and again this year, Bill will give a session on the individ-

ualized modules recently adopted in the revision.

Working on your behalf with this executive, Bill informs me, is rewarding. He is concerned about future financing of BCAMT endeavors. He well knows that rising costs in every sector eat away at the best of aspirations.

(PET PEEVE) Finding funds to support Kokoskin's drive to establish Vector as Canada's leading publication.

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List of Publisher's Representatives RESIDENT AND NON-RESIDENT

BILL KOKOSKIN

Tony .Vander Woude Addison-Wesley Canada Ltd. 444 Robson Street Vancouver, B.C. V6B 2B5 685-3622

Brian Sims D.C. Heath Canada Ltd. 4283 Arthur Drive Delta, B.C. V4K 2X1 946-6715

Chandler Hill Denoyer-Geppert 2808 Tilden Street West Seattle, Washington 98199 283-5071

Robert Wood Ginn & Company 8566 Terrace Drive North Delta, B.C. V4C 3Z2 581-8153

Lowell Smith Thomas Nelson & Sons 1007 - 1651 Harwood Street Vancouver, B.C. V6G 1Y2 687-6394

Raoul Bury McIntyre Educational Media 14431 Saturna Drive White Rock, B.C. V4B 3B6 531-7434

John Walker Visual Education 1869 West Georgia Street Vancouver, B.C. V6G 2W2 682-3788

Peter Clark Random House 15772 Goggs Avenue White Rock, B.C. V4B 2N7 536-9170

Doug Adams - Stu Calder Collier MacMillan Canada Ltd. P.O. Box 8780, Station H Vancouver, B.C. V6B 4E2

Jeff Stone MacMillan Co. of Canada Ltd. 130 - 19th Street West Suite 102 North Vancouver, B.C. V7M 1W4 980-2328

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Adrian Peetoom Scholastic Publications 5940 Canada Way Burnaby, B.C. V5E 3V7 526-1414

George Clarke - Paul Jones McGraw-Hill Ryerson Ltd. 1652 West 7th Avenue Vancouver, B.C. V6J 1S5 731-8013

Rick Pawliuk McGraw-Hill Ryerson (E.D.L. Div.) 104, 11460 - 40th Avenue Edmonton, Alberta T6J 0R5 434-0920

Dave Durbin Holt, Rinehart & Winston (Canada) Ltd.

11316 - 78th Avenue Delta, B.C. V4C 1N7 594-6044

Martin Goldberg McClelland & Stewart Ltd. 27592 - 110th Avenue, R.R.l Maple Ridge, B.C. V2X 7E6 526-1418

Bob Minor Readers Digest (Canada) Ltd. 3811 West 4th Avenue Vancouver, B.C. V6R 1P8 224-5893

Bob Skinner Educational Progress Co. 7265 Willingdon Avenue Burnaby, B.C. V5J 3R8 433-2127

Trevor Fox Van Nostrand Reinhold Ltd. 1595 East 57th Avenue Vancouver, B.C. V5P 2B2 327-0816

Sucha Gill Setsco Educational Ltd. 1315 West 71st Avenue Vanáouver, B.C. V6P 3B3 261-1361

Wes Edwards Gage Publishing Ltd. 600 Westview Centre North Vancouver, B.C. V7N 3X3 985-9367

John Morrow Chris Thomas Spectrum Educational Supplies Charles Merrill Canada Ltd. Ltd. 1102 - 1651 Harwood Street 315 - 1152 Mainland 3treet Vancouver, B.C. V6G 1Y2 Vancouver, B.C. V6B 2T9 681-1847 688-6522

Hank Luck Fitzhenry & Whiteside Ltd. Box 3287, MPO Vancouver, B.C. V6B 3X8 874-0511

Cam Millar Doubleday Canada Ltd. 202 - 828 Gilford Vancouver, B.C. V6G 2N6 682-0239

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Alan J. MacFarlane Howard Greaves Dent Edu-Media 1510 Bowser Avenue 5 - 27090 - 32nd Street North Vancouver, B.C. V7P 2Y3 Aldergrove, B.C. VOX lAO 987-8504 856-4289

Mike Wood G.L.C. 505 Spadina Street Chilliwack, B.C. V2P 1V3 792-8866

Don Greenwood Science Research Assoc. (Canada) Ltd.

403 Newdale Court North Vancouver, B.C. V7N 3H3 985-7680

Chris Jackson Oxford University Press 304 -4824 Georgia Street Delta, B.C. V4K 2S9 946-2047

Arthur Colbert Book Society of Canada Ltd. 5687 Wallace Street Vancouver, B.C. V6N 2A1 266-5138

Pat McGoey Houghton Mifflin Co. 1004 - 9500 Erickson Drive Burnaby, B.C. v3J 1M8 939-2551

Jack Hughes J.A. Hughes & Assoc. Ltd. 677 East 5th Avenue Vancouver, B.C. V5T 1H8 872-7217

Randy Koenig Talens C.A.C. Ltd. 1030 Leroi Street Vancouver, B.C. V5K 4G2 255-5378

Robert Blaney Ltd. 144 West 5th Avenue Vancouver, B.C. V5Y 1H9 872-8671

Steve Taggart Allan Stuart Copp Clark Publishing Sharp's Theatre Supplies 2918 Mathers Avenue 444 Coldfall Road West Vancouver, B.C. V7V 2K1 Richmond, B.C. V7C 1P8 926-6907 688-4832

Cameron Varcoe International Tele-film Enterprises 127 West 22nd Avenue Vancouver, B.C. V5Y 2G2 874-1436

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Nonresident

• Philip Lorenz Ms. Pat Foran Longmans (Canada) Ltd. Psycan Ltd. 55 Barber Greene Road 255 Consumers Road Don Mills, Ontario M3C 2A1 Willowdale, Ontario M2J 1R3 (416) 444-7331 (416) 491-4343

AN ACTIVITY APPROACH TO FRACTIONAL CONCEPTS

The Guidelines Committee of the Michigan Council of Teachers of Mathematics has released its fifth monograph, An Activity Approach to Fractional Concepts. The monograph deals with the teaching of fractional concepts, and the worksheets included are intended to be used as enrichment materials and as ideas for fur-ther conceptual development.

Copies of the publication are available for $1 each (prepaid) from Horace L. Mourer, MCTM Publications Chairman, 2165 E. Maple Road, Birmingham, Michigan 48008. Please make cheques payable to the Michigan Council of Teachers of Mathematics (MCTM).

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A Summer Institute in Computer Science

DAVID F. ROBITAILLE UNIVERSITY OF BRITISH COLUMBIA

If you know a computer programming language such as FORTRAN. or BASIC, we have a summer course for you. If you have Grade 11 or 12 students who display particular talents in computer science, we have a summer course for them too. The courses will be offered on the UBC campus under the auspices of the Centre for Continuing Education, beginning June 30, 1975 and lasting for three weeks. Fees for the courses are $50 for teachers and $25 for secondary school students. The organizers of this summer institute are John Peck, Head of the Computer Science Department at UBC, and I.

The course is intended to provide an introduction to some basic concepts in computer science. This is quite distinct from learning programming language. You cannot do computer science unless you know one or more programming languages, but learning programming is, in many cases, quite distinct from learning computer science. Also, certain programming languages are better vehicles for discussing computer science ideas than others. Because of this, many computer scientists would prefer not to teach languages such as BASIC or FORTRAN as 'a first programming language. They feel that these lan-guages are not among the best ones for the teaching of com-puter science.

Many people who are teaching computing at the secondary level have some course background in computer programming but little background in computer science. The summer institute is designed to deal with that problem. The course will treat such topics as computer architecture and algorithm

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construction. The former topic includes how computers work from a software (i.e., programming) point of view, while the latter- involves formulating procedures for computer-applica-

tion.

There will be two classes enrolled for the course: one consisting of teachers, and the other of Grade 11 and 12 students. Each section is limited to 25. Secondary school students must be recommended by their mathematics or com-puter science teachers. Classes will meet five days a week for three weeks, for approximately three hours a day. Each participant will be assigned a UBC computer account and will be expected to complete several computing assignments.

The course is noncredit. However, anyone who is interested in the course and who is a graduate student in Mathematics Education at UBC should discuss with the department's graduate advisor the possibility of using the course as a part of an independent study project.

Registration materials will be sent to all secondary schools in the-province by the Centre for Continuing Education. Should you require further information, contact Don Mosedale, of the Centre for Continuing Education (228-2181) or David Robitaille, Faculty of Education (228-5337).

CONSUMER EDUCATION KIT

Consumer Education Kit available free of charge to teachers from the B.C. Depart-ment of Consumer Services, 838 Fort Street, Victoria, B.C. V8W lH8. The kit contains samples of government brochures, copies of major acts, bibliographies of print and audio-visual materials, and a teacher's guide based on three films on advertising available from the Department of Education.

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Treasurer's Report 1974-1975 MARCH 25, TO MARCH 22, 1975

BCAMT

Credit $ 4,012.11 Balance - March 25, 1974

Receipts

Memberships 745 BCTF members 32 Non-BCTF members 46 Associate members

105 Student members BCTF Membership Grant Summer '74 Workshop (Final) 12th Northwest Conference (Final) BCTF Grant (Workshop Brochure) BCTF Grant (Executive Think Tank) Province of B.C. (Provincial Library) BCTF Grant

Disbursements

• $ '3580.00 320.00 230.00 105.00

3010.56 3356.32 773.40 429.62 300.00 30.00

429.6212,564.52

$ 16,576.63

Publications - four newsletter/journals $ 4161.31Third Summer Workshop (Surrey) 1498.48 12th Northwest Conference (Final) 192.44 Meetings Executive 1357.66

Committee 19.50 PSA Council assessment 78.40 AGM Expenses (1974) 329.90 NCTM affiliation fee 30.00 Executive Think Tank 678.00 Complimentary Card - Printing Cost 21.26 Summer Workshop '73 (Incorrect Posting) 91.76 Atlanta Conference Delegates Grant 50.00 B.C. Top Mathematics Scholar Awards 200.00 Membership overpayment refunds 22.00

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Disbursements (cont.)$ 16,576.63

Seed Monies - 4th Summer Workshop 500.00 BCAMT service pins 75.00 NCTM Leadership Conference 162.05 Photocopy, stationery, stamps, etc. 11.23 Election expenses 165.70 Constitution Printing Expenses 22.84 4th Summer Workshop (letters) 12.58

9,780.11

March 22, 1975 - Credit Balance $ 6,796.52

Accounts Receivable

Setsco Education Ltd. $ 210.11

Anticipated Expenses to June 30, 1975

Two newsletter/journals 2400.00 Three executive meetings 750.00 Aid to local chapters 300.00 AGM expenses 100.00 PSA Council Assessment 100.00 In--service Activities 200.00 Executive 'Think Tank' 400.00

$ 4250.00

Respectfully submitted to the BCAMT

W. Dale, Treasurer, BCAI4T

FREE METRIC MATERIAL

General Motors Corporation, Personnel Communications Dept., Personnel Administration and Development Staff, Detroit, Michigan 48202. Phone: 556-5000.

GMC has available a pamphlet entitled 'The Swing to Metric' which deals with the background, growth, principles, and impact of the metric system and its applications within GM. The pamphlet is suitable for Grades 6-12. (Multiple copies may be ordered.)

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A Book Worth Reading: 'WHY JOHNNY CAN'T ADD'

BARRY GEAR

Although Morris Kline exaggerates the use of principles in presumably teaching a student to add in his first chapter to entice the reader to proceed and does not sufficiently ex-plain 'Why Johnny Can't Add,' he does reveal problems and a possible shift in direction for math curriculum development.

Kline criticizes traditional and modern (particularly in-fluenced by the Cambridge Conference for School Mathematics) curricula and proposes criteria for selecting and sequencing topics, motivating students and evaluating a 'reformed' cur-riculum.

He feels traditional math contains many defects: a. Reliance upon memorization of processes and proofs. b. Disparate treatment of algebra and geometry. c. Minor logical defects. d. Retention of a few outmoded topics. e. Absence of motivational goal. f. Poorly written, commercially contrived materials.

Similarly, the new math, greatly influenced by rather narrow research mathematicians, contains many methodological prob-lems and some poor content.

The deductive approach, revered by the moderns is not ex-tolled by Kline - the attempt to promote understanding by this approach leaves out the intuitive probings necessary for students to understand new material. A deductive proof should come only when the students feel a need for it - the deductive approach should be the final step!

Kline fails to see the need for exacting rigor. At times,

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new texts assume what is vague and prove the obvious. He claims the capacity to appreciate rigor is a function of the mathematical age of the student and not the age (new) math.

Kline thinks there is an over-emphasis on language precision via definitions in new math. This is shown in several areas, including:

a. Number and numeral. b. Set language (does little to unify math as it is seldom used later in a course). C. Function as a relation in which no two number pairs have the same first number. d. Triangle and angle in some geometry texts, for example, merely result in, 200 terms to memorize in both Math 9 and 10 courses. Kline claims these definitions often conceal the scarcity of ideas and do not lead to clarity.

The emphasis on structure before the student has experience with numbers and operations hinders the overview which an appreciation of structure is supposed to give. The so called 'power' the structure approach is supposed to promote may be best developed by an exposure to math applications in real life.

Such content as other bases, modulo arithmetic, abstract algebra is generally meaningless if not preceded by effective learning of specific instances and applications. Perhaps they are poor examples of important math, also.

Kline claims the kernel of a math curriculum should stress breadth rather than depth. The content selected should transmit the following uses and values:

a. Math is the key to understanding the physical world. - b. Math gives us power over nature. C. Math gives us a conviction that we can continue to fathom the secrets of nature. d. Math is increasingly valuable to the social sciences. e. Math influences philosophy. f. Math is indispensable to technology.

Kline advocates that as a general teaching method, all sub-ject matter should be introduced intuitively - not deductively. This can. be accomplished via devices, pictures, heuristic

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arguments, inductive reasoning, reasoning by analogy and physical arguments. He suggests' that the math lab is a good method for teaching the above.

Topic sequencing should broadly follow the historical order of development, because it is usually, according to Kline, the most effective order for learning.

His evaluative criteria is simple - it is not how much math but to teach such that 50% of our students like and appreciate the significance of mathematics in our world.

MORE STATISTICS, FEWER HOURS

An ARITHEMTICAL STATIS-TICS PROJECT has been tried out in the New York City area. More topics are being cover-ed in less time by re-ordering calculations according to the arith-metical difficulty.

A complete set of print materials, including 2,000 worked-out pro-blems, is available for

reproduction at your institution. For these or for the corresponding videotape modules, write to Arithmetical Statistics Projects, c/o Richard G. Lefkon, Staten Island Community College, Staten Island, New York 10301.

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Problems for Computer Science

FRANK EBOS FACULTY OF EDUCATION

UNIVERSITY OF TORONTO

1. Find all the integers of three digits or less such that each integer is the sum of the cubes of its digits.

Example: 153 = + 5 3 + 33

2. Tabulate &, 2, x. x2 , x3, x, for l. x .Z 25.

3. Write a general function tabulator to write a table of values for any function of the form y = f(x)/g(x), given the first value of x, the change in x, and the final value of X.

Example: y = 2x - 3

as x goes frOm -5 to 5 in changes - 4x - 5

of 1.

4. Evaluate:

lx 2 + lx 2 x 3 + lx 2 x 3 x 4

for 1, 2, 3, 4, 5, ... 10 terms

5. Write a program to read values for a, b, c, d, e, f, and to solve the equations

ax + by = C dx + ey = f

The program should identify lines that are parallel, and lines that coincide as well as lines that intersect at a point.

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6. Write a program to read a list of numbers and determine which are prime.

7. List all the prime numbers less than 100 in an array five columns wide.

8. List the first 100 primes in an array five columns wide.

9. Find the highest common factor of two integers.

10. Read a number and find all the factors of the numbers.

Example: Factors of 12 are 1, 2, 3, 4, 6, 12.

11. Find all sets of integers that will form the sides of a right triangle. (Do not exceed an hypotenuse of 100.)

12. Find all sets of three integers that would form the length, width, and height measurements of a rectangle solid whose diagonal would also be an integer in length.

13. Read a number of marks, average them, and determine the number of marks in-each of the following intervals <50, 50 -- 59, 60 -- 65, 66 - 74, 75 -' 100.

14. Read a number of marks and calculate the mean and the standard deviation.

15. Read numbers from cards (eight per card in a F10.2 for-mat). Find the largest and the smallest number.

16. Read a series of integers between -10,000 and 10,000 and determine if each one is even or odd.

17. Find the prime factors of a list of integers.

Example: 36=2x2x3x3

18. Write a program to accept the scores of a series of games of a six-team hockey league and to produce a table of results under the headings - Team Name, Wins, Losses, Ties, Goals For, Goals Against, Points in Standing.

19. Write a program to calculate and update the hndicaps of a group of five bowlers (golfers).

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20. Write a program to calculate the point total and stand-ings of.a five team bowling league.

21'. Find the roots of a fifth degree equation with integer coefficients using the factor theorem.

22. Write-a program to find all n less than 100 such that the period of 1/n is equal to n - 1. Write n and the decimal equivalent of 1/n.

Example:1 =

23. A perfect number is a number that is equal to the sum of all its factors not including itself.

Example: 1+2+3=6

Write a program to identify and list all of the perfect numbers from one to 100.

The 4th Inter-American Conference on Mathematical Education shall take place in Caracas (Venezuela) from December 1 to 6, 1975. The enclosed circular describes the main objectives and the four themes of the conference, which is focused on mathematics teaching especially at elementary and secondary levels.

Those interested to receive further information are requested to contact

Claude Gaulin, Associate Professor Faculty of Education Université Laval Quebec City, Quebec G1K 7P4

Canadian representative on Comit Interamericano de Educación Matemtica

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Plan your vacation to include the

FOURTH MATHEMATICS SUMMER WORKSHOP Tuesday and Wednesday, August 26 and 27, 1975

New Westminster Secondary School

New Westminster, B.C.

PROGRAM HIGHLIGHTS:

60 Workshops (K-12) to choose from Repeats of many workshops Speakers from throughout B.C. Teaching mathematics in a multi-text environment

(K-12) Reaching curriculum objectives using the new.

textbooks (primary, intermediate, junior) Teaching the metric system (primary and inter-

mediate) Introducing the new Grade 9 and 10 textbooks The future for Math 11 and 12 Displays by over 20 publishers

Pre-registration $17 Students $10 On-site registration $19

PRE-REGISTRATION DEADLINE: JUNE 30, 1975

Registration fee includes a one year complimentary mem-bership in the B.C. Association of Mathematics Teachers and a subscription to its publication VECTOR. A PSA application form will be sent to you on receipt of your fee.

Registration fee, name, address, and telephone number should be included. Make cheques payable to: FOURTH MATHEMATICS SUMMER WORKSHOP and mail to:

FOURTH MATHEMATICS SUMMER WORKSHOP Box 69637, Station K Vancouver, B.C. V5K 4W7

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In-service Education: VIEWS OF TEACHERS

A Report of the Commission on the Education of Teachers of Mathematics*

A survey of opinions and perceptions of mathematics teachers concerning in-service education was conducted by the commission in the spring of 1974. Some of the results are reported below in the belief that you will find them useful and informative.

A sample of classroom teachers was selected by choosing each hundredth name on the NCTM membership roster and deleting all those names recognized as teachers at the collegiate level or workers in industry. Sixty-four percent of the sample responded. The results reported below are based upon 266 useable survey forms.

The typical respondent has tenure and 11 years of teaching experience, has course work or degrees beyond the bachelor's, and teaches in an urban or suburban junior or senior high school. Most (84%) of the sample were satisfied with the profession of teaching, and 87% would choose teaching as a career if they had it to do over again.

Most of the teachers sampled (83%) have participated in an in-service education program within the last four years. Generally, teachers have found in-service education benefi-cial, and 88% felt a need for in-service education. But 37% reported in-service education to have 'been a waste of time and energy,' and 61% have the perception that in-service education has not 'fit' classroom needs.

* Members: Ramona Choos, Helen Cunningham, Leroy Dalton, Virginia Gilbert, Alan Hoffer, Charles Hucka, Alan Osborne (Chairman), Alan Riedesel, Max Sobel, Irvin Vance; and former members, Ruth Greenwald and Viggo P. Hansen.

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The teachers were asked to list the three topics for which they felt the greatest need for in-service and, later in the form, to check in-service topics they felt would be most beneficial. In both question formats, the sample strongly favored methods over mathematical content. Topics relating to motivation received particular emphasis. Following are the topical preferences of these teachers. The number of teachers checking each topic is given.

Motivation 148 Geometry

Metrication 139 Career Education

Laboratory Learning 137 Open Education

Slow Learners 135 Discipline

Learning Styles Probability of Students 127

Topology Mathematical Games 125

Problems in Mathematics Development of Statistics Instructional Materials 124

Behavioral Objectives Problem Solving 112

Transformational Classroom Use of

Computers 112Geometry

Teaching Computation Applications 102

Cultural Characteristics Gifted Learners 100

of Students Enrichment 99

The Math I Am Teaching Individual Differences 96

Reading Consumer Math 95

Background on Math I Am Creativity 91 Teaching

Audio Visual 88 Algebra

Discovery Learning •87 Measurement

Evaluation 86 AP Calculus

Psychology of Learning 81 Proof

Testing 79 Calculus

Computer Languages 78 Coiribinatorics

Teachers showed no clear-cut preference for one structure

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70

66

64

61

56

53

51

50

49

48

44

41

40

39

37

35

31

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of in-service program over another. Single topic, two- to three-hour sessions were not significantly preferred to a series of meetings all concerned with the same in-service task. A small majority of the teachers preferred that in-service education programs be kept separate and distinct from university degree programs. The respondents were insistent that the control of in-service education programs should reside with the local school, in terms of both identifying in-service needs and designing the in-service program.

Teachers' opinions of mechanisms for encouraging in-service participation are indicated by the following questions. (The last number indicates the number omitting the question, the first number of percentage giving a yes answer.)

Should teachers be released from classroom duties for in-service? 75%, 15 Should teachers be paid expenses for in-service work rather than regard it as part of their professional responsibilities?

76%, 20

Parallel questions were asked about whether the teachers did in fact work in school systems that used these methods to encourage participation. The teachers who responded yes were generally much more positive about in-service work in respond-ing to all parts of the survey form. Indeed, those teachers who responded positively concerning their school's encouraging participation in in-service education tended to be positive in all aspects of their professionalism as revealed by the survey.

The strength of teacher beliefs in the payoff of in-service education is indicated by the response to these questions:

Should teachers participate in in-service education in order to maintain tenure? 57%, 20 Should in-service education be required of all who teach mathematics? 72%, 11 Should teachers participate in in-service education on a regular basis in order to maintain certification? 60%, 15 Should each school district have organized in-service activities and programs available for mathematics teachers?

90%, 8

Insight into the degree and type of administrative support for in-service was indicated by the responses to these

26

questions:

Does your school budget money for in-service? 57%, 32 If your school budgets money for in-service, is some of it reserved for programs about mathematics teaching and learning? 40%, 32 Is there an individual in your building or school system,

designated as responsible for in-service education? 39%, 24

If your school has an organized in-service program, do you participate in identifying the topics? 51%, 97

Given this optimistic, positive view of in-service education, one might expect that teachers are satisfied with in-service education as it is now conducted. Such is not the case. Eighty-three percent of the respondents to the question, 'Do you feel that there is a need for in-service efforts to be restructured?' responded yes. Almost a fifth of the teachers did not respond to the question. The large majority of the teachers who completed the open-ended portion of this ques-tion wrote concerning release time and problems of fitting in-service to their needs in the classroom. Many teachers who have had bad experiences with poorly planned and badly conducted in-service programs imposed on them with little consideration of their classroom needs used the open-ended questions to unload their perceptions in no uncertain term.

How typical are the responses of this sample of teachers in general? This is hard to judge. Clearly the respondents are a very professional group. They are atypical in that they belong to NCTM. But it is likely the perception of this professional group that in-service education needs restruc-turing would hold for less professionaly oriented teachers.

Annotated Metric Bibliography Compiled by: Shelagh M. Harris

Joan Routledge The York County Board of Education, October 1974

27

Highlights from Across the Country WHAT OTHER MATH ASSOCIATIONS ARE DOING

ED MURRIN

ALBERTA Held successful Annual Meeting at Jasper in October. George Cathcart re-elected president. Conducting metric workshops throughout the province. Has excellent publica-tions. Using NCTM grant to develop teacher-center.

SASKATCHEWAN - Planning a science-math symposium for fall 1975. Developing a' metric kit to be used for workshops. Participating in a department of education study on student evaluation. Continues to produce a fine publication but, like most associations, needs more contributors and additional financing. Jerry Gaski to co-ordinate metric workshop.

MANITOBA - Will host NCTM regional meeting (name-of-site) in October, 1976 at Winnipeg. Held outstanding workshop session in October 1974. There were about 1500 in 'attendance. It was my pleasure to be a participant. Three workshops planned throughout the province. Planning week-long seminar for May. Fine publication.

ONTARIO - Leadership seminar held in Niagara Falls in October. Next annual meeting scheduled for London in May. President John Egsgard is a nominee for president-elect of NCTM. We wish him the best of luck. OAME issues two fine publications, a newsletter and the gazette which provides the mathematical content. Plan to have a 'trading post' at future meetings for the exchange of materials - a novel and interesting idea. Association is deeply involved in curriculum development.

QUEBEC - The Province of Quebec has three mathematics teach-ers associations. Two French speaking, AMQ for secondary teachers and APAME for elementary teachers. QAMT is the

28

English-speaking association which provides for all levels.

AMQ and APAME produce excellent publications and conduct many fine workshops throughout the province. AMQ held its annual conference at Chicoutimi in October. APAME holds its major congress every two years. The next one is scheduled for May at Joliette. Several workshops are also held throughout the province during the year. QAMT held a successful workshop session at Ayers Cliff in September. Geometry was the topic.

NEW BRUNSWICK - Conducts a fall workshop each year in an area of the province distant from the location of the spring annual meeting. Annual meeting was held at Fredericton in November. It was my privilege to participate on behalf of NCTM. The math council is experiencing some difficulty in finding a new editor to replace Allison Kingston who has given long and dedicated service to the math teachers.

NOVA SCOTIA - New President, Mike Muise, who was elected at the annual meeting in October is a neighbor of mine in Antigonish. Lambertus Verberk remains editor and NCTM repre-sentative. A seminar was held last week to discuss the implications of the Graham Royal Commission recommendations on the mathematics curriculum and math teachers.

PRINCE EDWARD ISLAND - Several attempts were made to stimulate creation of an association in Prince Edward Island. No success as yet.

NEWFOUNDLAND - In addition to the annual meeting in May, which is held in St. John's, workshops are held in various areas of the province. Regional branches are also being established. The math council is very influential in curri-culum development and revision. Former President George Williams, is the new metric consultant for the department of education. Math council has matured rapidly, although only four years old.

If you have one lemma and add another lemma, you* have a dilemma.

29

Broad , Spectrum Mathematics Project

PETER WEYGANG ALAN C. MADGETT HEAD OF MATHEMATICS ASSISTANT PROFESSOR OF MATH LEVACK DISTRICT HIGH SCHOOL LAURENTIAN UNIVERSITY LEVACK, ONTARIO SUDBURY, ONTARIO

This project is an attempt to modify mathematics curricula and methodology at the senior high school and first-year university levels. We hope to create changes in mathematics education by putting into practice three fundamental ideas:

1. That a high school mathematics program should, for the majority of students, provide a wide but solid foundation. 2. That mathematics is best developed from a consideration of the realistic and relevant mathematics which occur in the everyday world of business, technology, and the pure and social sciences. 3. That the products of modern technology, namely the mini-calculator and the computer, are perfectly valid mathematical tools and should take their rightful place in mathematics education.

High school mathematics curricula have been controlled, to a large extent, by university entrance requirements. The presumption has always been that schools are preparing students for admission to the honors program in pure mathema-tics (which is not the intent of the average student). The majority of students are, in fact, proceeding to an arts program, life sciences, or other similar discipline in which the mathematics content is not of a highly theoretical or abstract nature, even though the amount of mathematics in-volvement and usage can be appreciable. Such students need a program which builds confidence by using simple mathemati-cal ideas, which builds flexibility by presenting a variety of mathematical alternatives, and which builds interest by

30

presenting material that is relevant to the real world.

The mathematics-user group has experienced a tremendous growth in recent years; it is difficult to find a profession, occupation or activity that does not have a mathematical component, which is often quite large. Even in baseball, batting averages are given to three-digit accuracy! In today's world, students are, from dire necessity, brought face to face with a mathematical reality. Many students are not 'turned on' by mathematics; as a result, these resentful students pose for the teacher a number of problems ranging from discipline to methodology. Motivation has become a big issue in mathematics education, because, unlike the classics, students cannot avoid mathematics without serious implications for later life.

Fortunately, most students are eager to get out into the real world, to become adults and full-fledged citizens. The school environment is artificial, and, by contrast, anything that smacks of realism has an inflated appeal. By using real problems as a starting point in mathematics, we do two things. We capture the interest of the student, by capital-izing on his desire for reality, and, we provide the students with a problem and a treatment in its original context, be it agriculture or nuclear medicine. This incidental back-ground information is of considerable value to the students, and it helps to enrich their vision of the world. This approach opens the students' eyes to career opportunities, provides some new insights, and, of course, teaches some good solid mathematic, s

It is apparent that students leaving high school enter a world in which the computer is commonplace. Every major university and industry has its own computer, every office desk has its mini-calculator. The age of 'speed of light' arithmetic has arrived everywhere, except in the mathematics classroom. Mathematics teachers have, in general, an in-grained distrust of the slide rule, calculator, or any other means of easing arithmetic computation. At the same time, they fail to appreciate that arithmetic difficulties are the prime hurdle to understanding mathematics. Computation is no friend to logic.

The study of fractions will illustrate this point. Histori-cally, fractions such as ½, ¼, 1/8 were associated with the

31

early tradesman in his day-to-day calculations, such as mak-ing window sashes, doors and the like. Manipulation of these fractions had a real value, since the alternative, reduction to decimals, was tedious. At some stage . , a mis-guided pedagogue decided that since 1/12 + 3/4 was a useful piece of mathematics, he was justified in producing lots of practice examples such as 5/7 + 4/11. This, in itself, was an error, since these fractions are almost impossible to conceptualize, as is the answer (83/77). Nonetheless, this fraction cult still dominates much of junior mathematics. If, for some curious reason, it were necessary to do this computation, the sensible way is to use the calculator:

5 4 + - = .71428571 + .36363636 = 1.07792207

7 11

This answer, probably close enough, took two seconds

Even at a higher level of mathematics, we should be using numerical methods and calculator aids. For example, numerical integration is a genuine alternative to theoretical integration. In fact, the problems that arise in industry are invariably represented by ill-behaved functions or purely empirical data, for which the slick methods of theoretical mathematics are seldom of much value.

Project History

The original concepts which underlie this project were for-mulated some five years ago by Peter Weygang, head of Mathematics at Levack District High School. This formulation was followed by a discussion on the project's acceptability with Ontario Ministry of Education representatives. John Milliken, now the Ministry of Education's liaison officer for this project, gave his approval in principle, and the way was clear for the next stage. This was a period of consultation with the mathematics faculty at Laurentian University, who provided a considerable amount of specific guidance as to desirable course content. Alan C. Madgett was a member of the original advisory group, as was J. Scott-Thomas, the project director.

Following this input, Peter Weygang developed a detailed course outline and some teaching units. This course, now in

32

the fourth year of in-class use, has surpassed expectations in all but one area - the number and variety of real life applications was too limited. The present cross-Canada sur-vey is an attempt to collect these materials.

Progress Report (As of September 30, 1974)

The collection of material started in British Columbia, where it was possible to enter into productive discussions with representatives of a wide variety of occupations such as oceanography, forestry, natural gas, and fisheries. Some classroom teaching was also done, in Kamloops, and the reac-tion to the approach and materials was very encouraging.

In Alberta, it has been possible to visit representatives of the plywood industry, corrosion engineers, petroleum engineers connected with the Athabaska Tar Sands, environ-mental specialists, and a host of others. In all areas, we have been able to obtain significant contributions, both as information and examples, which relate mathematics to real-world situations.

The collection stage of the project, which is funded by the Ontario Ministry of Education, will be completed in June 1975. At that time we will begin the mammoth task of col-lating and reworking the material into a form. that will be of maximum use to classroom teachers. The final form will probably be a set of resource books incorporating problems spanning a wide range of mathematical ability and grade le-vels. Each problem will be preceded by a short explanation of the technology involved, in order to 'set the scene.' The problem will be followed by a solution - sometimes several solutions of increasing refinement will be presented. The solutions will be followed by some notes and suggestions which may be of value to the teachers.

We are very fortunate to be able to undertake this project, for we appreciate that most teachers haven't the time, energy or financial resources to make such a survey. Even though most agree that the re-establishment of contacts between education and reality is one of our most pressing needs, we hope and trust that the end product of this project will be of use to all teachers of mathematics throughout Canada.

33

Sources of Funding

1. Ontario Ministry of Education: Grants in Aid of Educa-tional Research Program $20,000

2. P. Weygang: Ontario Secondary School Teachers Federation Traveling Scholarship $3,000

3. A.C. Madgett: President's Research Fund, Laurentian University $3,000

Visitations

British Columbia 1. Environment Canada, Vancouver (Commercial Fisheries

Operations) 2. Environment Canada, Patricia Bay, Victoria (Marine

Sciences Division) 3. Fisheries Research Board of Canada, Nanaimo 4. B.C. Forest Products Limited, Vancouver 5. Westcoast Transmission Limited, Vancouver (Natural Gas

Pipe Lines) 6. B.C. Hydro, Vancouver (Hydro-electric Power Generation) 7. Seaspan International Limited, Vancouver (Shipping) 8. BCIT, Industry Services Division, Vancouver

Alberta 1. Alberta Agriculture (Food Production), Edmonton 2. Alberta Environment, Edmonton (Air Management, Water

Quality, Hydrology, Pesticides) 3. Zeidler Plywood Corporation, Edmonton 4. Wardair Canada Limited, Edmonton (Air Transportation) 5. Caproco Corrosion Prevention Ltd., Edmonton 6. Gainers' Ltd., Edmonton (Meat Packers) 7. Great Canadian Oil Sands, Edmonton (Athabaska Oil Sands:

Oil Recovery) Athabaska Realty, Edmonton (Housing - Athabaska Oil Sands)

8. Numac Oil & Gas Ltd., Edmonton (Exploration) 9. Sherritt Gordon Mines Limited, Fort Saskatchewan (Nickel

Processing) 10. Alberta Research Council, Edmonton 11. Imperial Oil Gas Separation Plant, Redwater 12. Fleetwood Homes of Alberta, Red Deer 13. Energy Resources Conservation Board, Calgary 14. Canadian Petroleum Association, Calgary

34

15. Slumberger of Canada, Calgary 16. Dowell of Canada, Calgary 17. Water Survey of Canada, Calgary 18. City of Calgary: Transportation Department

Management Systems Development Department

19. General Hospital, Nuclear Medicine Dept., Edmonton 20. Sicks' Lethbridge Brewery, Lethbridge 21. Catelli Ltd., Lethbridge 22. Agri-Analysis Ltd., Lethbridge 23. Lilydale Poultry Sales, Lethbridge 24. Alberta Agriculture (Lethbridge): Poultry Branch

Each year, more mathematics teachers are realizing the relevance of the history or philosophy of mathematics to their own professional development and to their teaching.

This year, a new society has been formed to encourage teaching, study, and research in the history and philosophy of mathematics throughout Canada.

Founded in June 1974, the society will hold annual meetings in Canada. In the past six months, we have. grown to our present membership of 100, and we hope soon to publish a list of members and their areas of interest.

If you have an interest in the history or philosophy of mathematics, we invite you to join by requesting an application form. Write to:

CAN. SOC. HIST. & PHIL. OF MATHS SOC. CAN. WHIST. & DE PHIL. DES MATH. SIMON FRASER UNIVERSITY 8888 BARNETT HIGHWAY NORTH BURNABY, B.C. V5A 1S6

35

Preliminary Drafts of Proposed Courses for.Years 11 and 12

MATH REVISION COMMITTEE

In consultation with many teachers this preliminary draft has been revised many times and it is expected that more revision will take place after your comments have been received.

The Math Revision Committee would like your written comments, proposals, etc., of the following drafts for Mathematics at the years 11 and 12 level.

Send your comments to:

Math Revision Committee do Will Dunlop Curriculum Consultant Department of Education Parliament Buildings Victoria, B.C.

UBC has openings for two-three elementary or secondary teachers to apply for a one year term at UBC, Faculty of Education. They will teach the content and methods of math at UBC.

Write to: Ian Beattie Mathematics Education Faculty of Education UBC 2075 Wesbrook Place Vancouver, B.C. V6T 1W5 (228-5204)

36

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Distributor Cap Mathematics

REPRINTED FROM DELTA-K, VOLUME XIII, NUMBER 3, MAY 1974 THE ALBERTA TEACHERS MATHEMATICS COUNCIL

WILLIAM J . BRUCE UNIVERSITY OF ALBERTA

EDMONTON, ALBERTA

It is interesting to watch a skilled mechanic install a new distributor cap in a modern automobile. How quickly he connects the spark plug wires It appears that he plugs the ends into the distributor cap connectors in any convenient order, but this is not really what he does. There is one and only one correct arrangement that will work. How does he know the correct one? Ask any experienced mechanic, and he'll probably tell you.

Let us suppose that we don't know the trick and that we start plugging in at random. How many arrangements are possible? The standard Delco-Remy distributor cap for an eight-cylinder car is shown in Figure 1. We note that the connectors for the eight spark plug wires are arranged in a circle and that there is a center connector for the power source. At the base are two clip mechanisms for fastening the cap. We shall examine the problem for motors with different numbers of spark plug wires. Figure 2 is a diagrammatic representation of a distributor cap for a four-cylinder motor. The spark plug wire connectors are shown as 1, 2, 3, and 4, 'c' is the center connector and 'k' is a clip. It must be noted that, once installed, the cap is rigid. We consider the number of possible ways to plug in the spark plug wires. Suppose that the first wire that we choose is plugged into position 1. Then there are 3! ways of plugging in the other three wires for each position of the first wire. But the first wire could have been plugged in at four different places. Thus, there are 4(3) or 4 possible arrangements, a total of 24

49

ways to plug in the four wires.

0 Figure 1 Figure 2

0oo®.

Consider, now, caps for six- and eight-cylinder motors as diagrammed in Figures 3 and 4. For the six-cylinder motor, we plug into position 1 again and permute the other five to obtain 5 ways of plugging in the other five wires.

Figure 3 Figure 4

If we allow the first wire to be plugged into any of the six

places, we obtain 6(5) or 6; that is, 720 ways to plug in the six wires. Proceeding in the same way for the eight-cylinder motor, we find that there are 8(7:) or 8; that is, 40,320 ways to plug in the eight wires - obviously far too many for a trial-and-error approach.

Suppose further that the wire to the center connector is in-cluded with the spark plug wires and that all arrangements are determined with center wire and spark plug wires con-sidered interchangeable. From Figure 2, we see that there are now 5 ways of plugging into the center position and for each of these there are 4 ways of plugging into the spark plug

50

connectors, a total of 5(4!) or 5!, that is, 120 ways in all. From Figure 3, we similarly obtain 7 or 5040 ways. From Figure 4 we get 91 or 362,880 ways.

Remove the clip pins from a Delco-Remy distributor cap and use it as a pencil or crayon holder for your desk. If pre-ferred, the black color can be changed to suit any decor by spray painting. An examination of Figures 2, 3, and 4 re-veals that we have, respectively, 7, 9, and 11 places for pencils when we in-clude the clip positions 'k.'

Suppose that all positions, except the center position, are to be filled with different colored pencils and that we do not allow the holder to rotate. Consider the number of possible arrangements of the pencils in each case, exiuding the center position. From Figure 2, we see that there are 6(5) ways of filling positions 'k,' and for each of these there are 41 ways of filling positions 1, 2, 3, and 4. Hence there are 6(5)(4!) or 6!; that is, 720 ways of arrang-ing the pencils. From Figures 3 and 4, we find similarly that there are 8! (that is, 40,320) and 10! (that is, 3628,800) ways, respectively.

If we include the center position 'c' in Figure 2, we find that there are 7(6) (5) ways of filling positions 'k' and 'c' together, and for each of these there are 4 ways of filling positions 1, 2, 3, and 4. This yields a total of 7(6) (5) (4!) or 7!; that is, 5040 ways of arranging the pencils. From Figures 3 and 4, we find similarly that there are 9! (that is, 362,880) and 11! (that is, 39,916,800) ways, respectively.

The following table summarizes our results. If rotations had been permftted, the cyclic permutations would have been computed using (n - l) instead of n! as has been done.

n p p + c p+2k p+c+2k

4 4! = 24 5! = 120 6! = 720 7! = 5040

6 6! = 720 7! = 5040 8! = 40,320 9! = 362,880

8 81 = 40,320 9! = 362,880 10! = 3,628,800 11! = 39,916,800

n - number of spark plug wires.

51

p - number of cyclic permutations of the spark plug wire connections to the distributor cap (including rotations of the wires).

c - center connection on the distributor cap. k - clip points of the distributor cap.

EARLY LEARNING YEARBOOK RELEASED

MATHEMATICS LEARNING IN EARLY CHILDHOOD is NCTM's colorful and highly illustrated 37th V Yearbook, edited by Joseph Payne of the University of Michigan and designed for teachers of children aged three to eight. A general framework built on the current knowledge of children's cognitive develop-ment, of the way they learn mathematics, and of curriculum design is followed by eight chapters dealing with the learning and teaching of specific mathematical content. Activities designed to engage children, to elicit thoughtful responses, and to produce effective learning permeate these 'content' chap-ters. Special curriculum projects and directions of curricular change are discussed in the final chapter.

The 30O-page book stresses three themes: problem solv-ing, how to relate mathematics to the real world of the child, and the developmental nature of learning. A valuable source of both knowledge and activities for any-one who works with young children, the 37th Yearbook is hardbound and printed in the oversized, informal format. The many practical suggestions should appeal to the classroom teacher, and the research theoretical work will be useful to researchers and scholars in mathematics education. The book may be purchased from the NCTM Headquarters Office for $12. NCTM members are entitled to one copy at $11.

52

Games People Play A STARTING LIST OF 'MATH' GAMES FOR ELEMENTARY SCHOOL

FRANK EBOS UNIVERSITY OF TORONTO

NUMBER GAMES

NAME COMMENTS SOURCE +

NUMBLE MULTIPLES OF 3 T

QUINTO MULTIPLES of 5 1 G

SUM-TIMES MULTIPLES OF SJERALSCRABBLE

3M TYPE

NUMBERS

WINNINGMULTIPLICATION FACTS M

TOUCH

YA}ffZE C DICE-POKER

YUM P

STOCK BASIC OPERATIONS IN A BUSINESS TICKER CONTEXT C

HEADS UP EQUATIONS BASED ON BASIC C j OPERATIONS

TUF E

ORDOCARDS M-7 (KRYPTO) ORDER OF

OPERATIONS RECKON DICE IN VARIOUS SHAPES GW

MATH RUMMY NUMERALS FOR THE SAME NUMBER M-7

+ SEE LAST PAGE FOR FULL LISTING

53

NUMBER GAMES (cont.

NAME COMMENTS SOURCE

MATHPOLY BASIC OPERATIONS M-5

MATH'N'-VARIABLE CONTENT POSSIBLE M-7

TRATION

LAS EQUIVALENT FRACTIONS - BETTINGH

FRACTAS FORMAT

WHAT'SFRACTIONS H/OMC

BELOW

FRACTIONEQUIVALENT FRACTIONS M

DOMINOES

MATH PORT H

MAN H OVERBOARD

INTEGERS INTEGER H/OMC ARITHMETIC

INTEGRAD M-6

IMMA WHIZ ADD'N, MULT. M

SMARTY ADD-N, SUBT'N M BINGO

ADDO, TYPEM

MULTO ADD'N, SUBT'N

NUMO SEVERAL LEVELS J . Mi

RAKO7STRATEGY

Mi IN

NUMERO ORDERING . H NUMBERS

TOTAL OUT 3M

POWER PLAY HOCKEY CONTEXT H POWERS

} SWIM MEET SWIMMING CONTEXT H

PRIME DRAG PRIME, COMPOSITE NUMBERS CR

BASIS PLACE VALUE GW

MAGIC PRIMARY NUMERAL, SHAPE, H NUMBER COLOR RECOGNITION

54

NUMBER GAMES (con t.)


BINARYBINARY NUMERATION SYSTEM

DOMINOES M

UPS AND BASIC OPERATIONS SNAKES ANDH

DOWNS AND LADDERS FORMAT

SKEDOODLE PAPER & PENCIL BASIC OPERATIONSGG

SOURCE

HEKATON CARD GAME, TRICKS TAKEN ONGG

BASIS OF COMPUTATION

GEOMETRY, MEASUREMENT GAMES


MATH ANGLE CONSTRUCTION, RATIO ANDM-6

LINKS PROPORTION

LAVD MEASUREMENT, BINGO TYPE M-7

SYMMETRYPRIMARY GEOMETRIC SHAPES M

DOMINOES

SHAPE-0 FITTING SHAPES INTO A CONTAINER T

SIM BASED ON A HEXAGON H

HIP AVOID MAKING SQUARES H

TAKITOR-STRATEGY BASED ON SYMMETRY MHE

LEVIT

SPIN ANDPRIMARY SHAPE RECOGNITION PL

MATCH

BUILD A HOUSE PRIMARY SHAPE, NUMBER, MONEY PL GAME

55

CO-ORDINATES, TRANSFORMATION GAMES

NAME / COMMENTS SOURCE

CO-ORDINATE INTEGER CO-ORDINATES H TIC-TAC-TOE

BATTLESHIP INTRO TO CO-ORDINATES M-6

VECTORAMA CO-ORDINATE GRID TRANSLATIONS H

VECTOR GW

PATHFINDER H

TRANSFRA- TRANSLATIONS H TION

MIRROR REFLECTIONS MA MANIA

RACE TRACK AUTO RACE SIMULATION SA

'L' GAME TRANSFORMATIONS H

PICK THE H PENTOMINOES

TIC- TA C- TOE- TYPE GAMES


CUBIT 3D-COLORED CUBES Mc

QUBIC 3D-4x4x4 GAME BOARD M

TAC TICKLE WFF'N PROOF SERIES G

GUNG HO CLASSIC CHINESE GAME C

BALANCED CENTER POSITION HASH

TIC-TAC-TOE DISAPPEARED

CUBE FUSION EVERY MOVE IS A MOVE FOR YOURHS OPPONENT

LOGI-QUAD 3D-PYRAMID FORM GW

56

BINARY BASED STRATEGY GAMES


DOCTOR NIM M MAN VS COMPUTER THINK A

DOT M

AVALANCHE LOTS OF ACTION P

HEXAPAWN COMPUTER BASED 0 IBM

NIM PAPER VERSION OF DR. NIM H

CHESS/CHECKERS TYPE GAMES


CROWN RED RANDOM CHESSLIKE MOVES P

SPACECHECKERS IN 3-D PA CHECKERS

SMESS CHESS FOR 'NINNYS' P

ATTRIBUTES GAMES


DIFFERENCELOGIC BLOCKS SC GAME

MATHEMAX CARDS - VARIETY OF GAMES

ATTRIBUTES, GAMES AND KIT: VARIETY OF ACTIVITIES Mc PROBLEMS

57

MISCELLANEOUS GAMES

NAME. - COMMENTS SOURCE +

KALAH 'I A }ANCIENT ARABIAN GAME

MANCAILA M

HEXOGRAMS GAME OF HEX: COLOR MATCHING M

CONNECT T

PATTERN MATCHING HAAR

)GW

HOOLIM

DEAD DOT LINE DRAWING COMPETITION H

HEX MAKE A LINE ACROSS A HEXAGON m-7 BOARD

TWIXT BUILD BRIDGES ACROSS A GRID 3M

FOOTBALL

• SOCCER FORMAT ON A GRID H ON PAPER

FRENCH MILITARY STRATEGY-POSITIONAL GAME • H/MG

GAME

THREE ARRANGING PIECES ON CHECKERBOARD GG MUSKETEERS

PUZZLES


JOIN THELINE DRAWING PUZZLE H

SQUARESCR

TRIGRAMSTANGRAMS CR

MADAGASCAR MADNESS

MULTIPUZZLE HEXOMINOES M

TOWER OF H HANOI

CLASSIC PUZZLE

58

PUZZLES (cont.)


LATIN 'ATTRIBUTES PROBLEMS AND GAMES' F SQUARES KIT Mc

BALI BUTTONS LATIN SQUARE FORMAT GW

INSTANTPOPULAR 4-COLOR PUZZLE P INSANITY

SOMAREASSEMBLING A CUBE IMPUZZA-

BLE CR

MATH JUMP PATTERN ON A GRID H

SWITCH'EM REVERSING POSITIONS OF PEGS H

PHONEY BALONEY

GEOMETRY PUZZLE CHALLENGE DICE GW

DOUBLE TROUBLE COLOR MATCHING GW

GAMES BIBLIOGRAPHIES

1. GAMES AND PUZZLES: MONTHLY BRITISH PERIODICAL FOUNDED IN 1972. (P.O. BOX 4, LONDON N6 4DF)

2. STRATEGY AND TACTICS: BIMONTHLY PERIODICAL WITH A COLUMN BY SIDNEY SACKSON ON THE LATEST MATHEMATICAL GAMES. (44 EAST 23RD STREET NEW YORK, N.Y. 10010)

3. THE ARITHMETIC TEACHER: NCTM MONTHLY PERIODICAL WITH A SECTION ON NEW MATHEMATICAL

59

GAMES BIBLIOGRAPHIES (cont.)

GAMES (1201 16TH STREET N.W. WASHINGTON, D.C. 20036)

4. SCIENTIFIC AMERICAN: MONTHLY PUBLICATION WITH COLUMN ON MATHEMATICAL GAMES BY MARTIN GARDNER (415 MADISON AVENUE NEW YORK, N.Y. 10017)

5. A GAMUT OF GAMES: S. SACKSON, RANDOM HOUSE, 1969 CONTAINS A BIBLIOGRAPHY OF MORE THAN 200 OF THE BEST MATHEMATICAL BOARD GAMES ON THE MARKET

6. MATHACTION 1-7: (8 BEING PUBLISHED), BATES, ET AL., COPP CLARK A CONSIDERABLE VARIETY OF GAMES FOR ALL ELEMENTARY LEVELS (ALL TRIED IN CLASSROOM SITUATIONS)

SOURCES

A Acme Ruler Company E Educational Supply Co. 600 Victoria Park Avenue 105 Brisbane Road Toronto, Ontario M4E 3T6 Downsview, Ontario M3J 2K7

Arrco- Saxon Company 3195 Airway Drive Malton, Ontario L4V 1C2 677-8030 (Blank Playing Cards)

C Copp Clark Publishing Company 517 Wellington Street West Toronto, Ontario M5V 1G1

CR Creative Publications P.O. Box 10328 Palo Alto, California 94303

G Games For Thinkers Company 1111 Finch Avenue West Downsview, Ontario M3J 2E5

GA Mathematics Book 1 Gage Publishing Company 164 Commander Blvd. Agincourt, Ontario M1S 3C7

GG A Gamut of Games S. Sackson Random House, 1969

60

SOURCES (cont.)

GW Gameways Mi Midwest Publications Co. 91 Bloor Street West P.O. Box 307 Toronto, Ontario M5S 1M1 Birmingham, Michigan 48012

H Homemade, often by 3M Minnesota, Mining and Frank Ebos' University of Manufacturing Toronto Students 1415 Lawrence Avenue West (original or drawn from Toronto, Ontario M6L 1A9 various sources)

OMC Ontario Mathematics HS House of Games Corporation Commission

801 York Mills Road Toronto, Ontario M3B 1X7 P Parker Brothers Games Ltd.

122 Cartwright Avenue M Moyer Division Toronto, Ontario M6A 1V2

Vilas Industries 25 Milvan Drive Weston, Ontario M9L 1Z1

M-5, M-6, M-7 Mathaction 5, Mathaction 6, Mathaction 7; Bates, et al. - Copp Clark

Toytown Co. Ltd. 4362 Chesswood Drive

MG Martin Gardner's Toronto, Ontario M3J 2B9 Sixth Book of Mathematical Games from Scientific PL Playskool Game American Louise Kool & Son Ltd.

132 Sunrise Mc McGraw Hill Co. of Canada Toronto, Ontario M4A 1B6

330 Progress Avenue Scarborough, Ontario M1P 2Z5

MHE Mathematics - A Human Endeavour H. Jacobs W.H. Freeman Co., 1970

PA Pacific Game Company North Hollywood, California

SA Scientific American Magazine, January 1973

SC Scholar's Choice Stratford Division 50 Ballantyne Avenue

MA Mattel Canada Limited Stratford, Ontario NSA 3A5 760 Lawrence Avenue West Toronto, Ontario M6A 1B8 T

61

Games People Play A STARTING LIST OF 'MATH' GAMES FOR SECONDARY SCHOOL

FRANK EBOS UNIVERSITY OF TORONTO

NUMBER GAMES

SOURCE NAME COMMENTS

T NUMBLESCRABBLE-TYPE, NATURAL NUMBERS

G QUINTO J C YAHTZE DICE-POKER, NATURAL NUMBERS

C STOCK BASIC OPERATIONS IN TICKER BUSINESS CONTEXT

C HEADS UP

H/M-7ORDER OF OPERATIONS,

(KRYPTO)NATURAL NUMBERS

GW RECKON

MI RANKO

M WINNING MULTIPLICATION FACTS: COULD TOUCH EXTEND BEYOND NATURAL NUMBERS

H/M-7 MATH N TRATION

FRACTIONS H LAS FRACTAS

H/OMC WHAT'S BELOW

62

NUMBER GAMES (cont.)


H MATH PORT

H MAN OVERBOARD INTEGERS

H/OMC INTEGER ARITHMETIC

E TUF

G WFF'N OPERATIONS AND EQUATIONS PROOF - IN A VARIETY OF NUMBER EQUATIONS SYSTEMS

G REAL NUMBERS

ALGEBRA GAMES

NOTE: MOST OF THESE GAMES COULD BE USED TO DRILL ANY NUMERIC OR ALGEBRAIC IDEAS


H FUMMY RUMMY TYPEOF FUNCTIONS H ALGEBRIDGE BRIDGE TYPE

H QUAD-RAT-PROPERTIES OF QUADRATIC FUNCTION IT

H FACTO BINGO TYPE

H MATCHING } FACTORING

H RACE LINE ANALYTIC GEOMETRY - CARD GAME

H/OMG BASEBALL ALGEBRA REVIEW

H POWER PLAY} EXPONENTS

H SWIM MEET

63

CO-ORDINATES, VECTORS, TRANSFORMATIONS GAMES


H/GA BATTLESHIPTO CO-ORDINATES

H CO-ORDINATEfINTRODUCTION

TIC-TAC-TOE

VECTORAMA CO-ORDINATES, VECTORS

H PATHFINDERVECTORS, TrthSLATIOWS

GW VECTOR

H TRANSFRATION TRANSLATIONS

MA MIRROR MANIA REFLECTIONS

GW HAAR HOOLIM

H 'L' GAMETRANSFORMATIONS

H PICK THE PENTOMINOES

GEOMETRY GAMES


H/M-6 GOLF ANGLE CONSTRUCTION, RATIO & PROPORTION

MI EUCLID BASIC DEDUCTIONS

H/M-7 LAVD BINGO TYPE

MISCELLANEOUS


H LOKER}LOGIC

G WFF

64

MISCELLANEOUS (cont.)


G WFF'N LOGIC PROOF

G ONSETS

H TRIGGER TRIG, IDENTITIES

STRATEGY GAMES


H CHECKER-ETTES (TAC TICKLE)

2-D TIC-TAC-TOE H BALANCED

TIC-TAC-TOE

CHECKRO COMBINATION OF 84 & 49

MA TURNING 2-D TIC-TAC-TOE TYPE; POINT CAPTURING CHIPS

C GUNG HO 2-D TIC-TAC-TOE TYPE; 5 IN A ROW

M-7 HEX 2-D TIC-TAC-TOE TYPE; MANY IN A ROW

M QUBIC

• HS CUBE • FUSION 3-D TIC-TAC-TOE

H 3-D TIC-TAC-TOE

GW LOGI-QUAD PLACING MARBLES IN A PYRAMID IN CERTAIN ORDERS

M' DR. NIM1BINARY BASED GAMES

H NIM J

65

STRATEGY GAMES (cont.)


M THINK DOT IJBINARY BASED GAMES

P AVALANCHE

H DEAD DOT

H FOOTBALL ON PAPER

IBM HEXAPAWN CAPTURING OPPONENTS ON A 3 x 3 BOARD

CHESS

.P SMESS 1SIMPLER THAN CHESS BUT JCHESS-LIKE MOVES

P CROWN RED

CHECKERS

PA 3-D CHECKERS CHECKERS

H/MG FRENCH MILITARY GAME

A KALAH (MADAGASCAR)

SOURCES

A Acme Ruler Company C Copp Clark Publishing 600 Victoria Park Avenue Company Toronto, Ontario .M4E 3T6 517 WellingtonStreet West

Toronto, Ontario M5V 1G1 Arrco- Saxon Company 3195 Airway Drive Malton, Ontario L4V 1C2 677-8030 (Blank Playing Cards)

G Games For Thinkers Company 1111 Finch Avenue West Downsview, Ontario M3J 2E5

66

SOURCE (cont.)

GA Mathematics Book 1 MG Martin Gardner's Gage Publishing Company Sixth Book of Mathematical 164 Commander Blvd. Games from Scientific Agincourt, Ontario MiS 3C7 American

GW Gameways MI Midwest Publications Co. 91 Bloor Street West P.O. Box 307 Toronto, Ontario M5S lMl Birmingham, Michigan 48012

H Homemade, often by Frank OMC Ontario Mathematics Ebos' University of Commission Toronto Students

0MG Ontario Mathematics HS House of Games Corporation Gazette, September 1972

801 York Mills Road Toronto, Ontario M3B 1X7 P Parker Brothers 'Games Ltd.

122 Cartwright Avenue M Moyer Division Toronto, Ontario M6A lV2

Vilas Industries 25 Milvan Drive PA Pacific Game Company Weston, Ontario M9L 1Z1 North Hollywood, California

M-5, M-6, M-7 Mathaction 5, Mathaction 6, Mathaction 7; Bates, et al.-- Copp Clark

MA Mattel Canada Limited 760 Lawrence Avenue West Toronto, Ontario M6A 1B8

T Toytown Co. Ltd. 4362 Chesswood Drive Toronto, Ontario M3J 2B9

67

Experiences With Functions

Reprinted with permission from the New York State Mathematics Teachers' Journal,' Volume 23, Number 3, June 1973.

J.B. HARKEN STATE UNIVERSITY COLLEGE

OF BROCKPORT

Introduction

Functions pervade all of mathematics. Functions play a role as prominent as that of numbers and are emerging explicitly in the elementary school mathematics curriculum. Functions and numbers are, so to speak, the 'bread and butter' of mathematics.

In searching for a rule to relate sets of numeral or physical objects, we are usually looking for a function. Before exploring some embodiments of numerical functions, we will survey the development of the concept. Embodiments provide an experiential base on which to build the abstract concept of a function. An embodiment, by definition, is an ex-perience that realizes the concept. It provides an instance in which the concept is interpreted. Some of the embodiments are provided in a discovery situation that leads to the formation of a rule or function. Other embodiments simply require that we explicitly formulate a rule to accompany an already identified geometric equivalent of the function.

Procedures for combining functions to obtain other functions suggest that functions are elements of a system [1]. Opera-tions with functions are more complex than those of arithme-tic. Not only are the function operations of addition and multiplication more sophisticated than their analogs in number systems, but also functions generate a new operation called composition or substitution. Composition of functions

68

has no analog in number systems.

We are going to restrict our development of embodiments of functions to those functions related exclusively to the real number system. These functions form a set described techni-cally by the phrase 'real, numerical functions.'

History of the Concept of Function

There are three discernible major stages in the 'evolution of the first explicit concept of function. In the work of the Greek mathematicians, we find the germ of the idea. Loci-plotting implicitly contains the ideas of variable and func-tion. Greek considerations of problems of area and curvature would ultimately have led to the reciprocity laws of the calculus. It was the need for consistency in the calculus that eventually required a first and explicit clarification of the concept of the function.

While the function concept was in an implicit stage, contri-butions to symbolism in mathematics were made by Vieta. His study of equations was the first major development that pro-vided a symbolic framework, which helped initiate the ex-plicit appearance of the concept of function.

A second major development occurred in the field of kinema-tics. Oresme used a graphical representation to show that the velocity of a freely falling body is proportional to time. Galileo formulated laws for rectilinear motion and falling bodies.

Analytic geometry became the meeting place for the curves resulting from the study of equations and the motion studies of Oresme and Galileo. Curves at the time of Descartes were strictly of an algebraic nature. Nonalgebraic curves were later encountered by Barrow and Newton. Barrow's interest lay in smooth (continuous) curves. Newton's use of the idea of 'fluent' brought him very close to the notion of a numeri -cal function. A 'fluent' is a consistent class of related quantities, one of which is not numerical.

J. Bernoulli formulated the first explicit definition of a function in 1718. Leibnitz coined the word 'function.' Functions are 'quantities formed in any manner whatever of

69

ZVV Sawtooth curve

an independent variable and constants.' An explicit for-mula involving variables is required in this concept. In the mathematics of that period the following graph in Figure 1 would not represent a function since two rules are needed to describe the graph.

Figure 1 Figure 2

Function notation was introduced by Clairaut in 1734. The first treatise on algebraic and transcendental functions was that of Euler. He freed the function concept from geometry, introduced implicit functions, and made a distinction be-tween single-valued and many-valued functions.

Euler's work stimulated some disturbing examples of curves outside the scope of the Bernoulli conception. Fourier found it necessary to formulate the second, and broader, con-cept of a function: 'We must admit into analysis functions which have equal values, whenever the variable receives any values whatever included between two given limits ....' In the Fourier sense, any curve you can draw is a function. Thus, sawtooth curves (Figure 2) though not Bernoulli func-tions, are Fourier functions. Outside the scope of the Fourier function is the set of pairs that cannot be graphed. For instance, the characteristic function of the rationals,

11, if x is a rational f(x) = I 0, if x is not rational

is not within the reach of the Fourier concept.

Our modern concept of a function is that of Dirichlet (1829). This concept is the synthesis of the Bernoulli and Fourier concepts and the demands imposed by need to consider such numerical objects as the characteristic function of the rationals. In Dirichlet'S words: 'Let a and b be two fixed values, and let x be of variable size so that x takes on, little by little, all values between a and b. If a unique finite y corresponds, to each x so that, while x traverses the

70

interval from a to b continuously, y = f(x) similarly changes gradually, then y is called a regular function of x over this interval.' Within the scope of this definition are Bolzano's continuous and everywhere-changing-direction curve as well as the continuous space-filling (two dimensional) curve of Peano. In our present conception we no longer require the regularity (smoothness) imposed by Dirichlet.

E'nbodiments of the Function Concept

Functions can be classified as 1-place, 2-place, or n-place. Before formalizing the notion of function and a system for the classification of functions, experience with specific embodiments of the concept is a fruitful psychological foundation. An interlacing of embodiments and abstractions provides a rich cognitive framework for the growth of the concept. Consistent confrontations between abstractions and embodiments heighten the meaningfulness of the concept. In-tellectual integration is improved as the disparity between experiences and symbols is diminished.

One-Place Function Embodiments

Most of our 1-place embodiments have their origins in physics. Some are in geometry, others in growth patterns. The funda-mental objects of concern in physics are semi-conceptual entities called fluents. Distance, time, volume, temperature, etc. are examples of fluents. A fluent spans both the real and the conceptual worlds. For instance, distance is characterized by the following pair: (act of measuring with a metre stick, number result of the act of measuring). With each physical act of measurement on a calibrated instrument there is associated a conceptual object - a number.

A fluent is a class of pairs in which the first element in a pair is an act of measuring on an instrument and the second element is the corresponding number determined by the instru-ment.

Functions serve as connecting links between fluents. In each of the embodiments that follow the exploration leads to a rule - a function.

71

Peg Game

Although the peg game comes in various disguises, the form used consists of eight golf tees (4 white and 4 yellow) and a wooden holder with nine holes.

The rules for the game are:

1. The white pegs may move only to the right, the yellow pegs only to the left. 2. You may move only one peg at a time. 3. You may move a peg into an adjacent hole. 4. You may jump, but only a single peg of the opposite color. 5. To win the game all white pegs must be replaced by yellow pegs and all yellow by white.

Data: Number of pegs Number of moves to on one side interchange (minimum)

o A

1 3

2

3

4

When the table has been completed, draw a graph and discover the rule.

Tower Puzzle

Materials needed are six tiers (discs) with openings in cen-ters of each and a three tower board.

Object: Transfer tiers in minimum moves from one tower to another tower.

Rules: Move one disc at a time.

72

Cannot place a larger disc on a smaller disc.

Data: Number of tiers Number of moves to transfer (minimum)

0

1 1

2 3

3

4

5

6

94

When the table has been completed, draw a graph and discover the rule.

Two-Place Function Embodiments: Pick's Formula

This experience is constructed for you to discover Pick's formula for the area of a simple closed polygon. This rule is an embodiment of a two-place function that is within the scope of elementary school mathematics.

As a first experience, consider the following twelve pro-blems. Adjacent to each problem are the placeholders 0 and A. The instructions at the top of the figure require that you count the grid points that lie on the simple closed polygon and record the number in the 0 . The number of points in the interior of the simple closed polygon is re-corded in A.

73

After finishing this counting chore, proceed to the partially completed table. While filling in the missing entries, probe for a pattern and formulate it as a rule. This rule should tell you how to combine the D and A points to calculate the area, S, of any simple closed polygon.

There are empty spaces provided at the bottom of the table. If a pattern doesn't emerge in the twelve problems provided, create more problems and record more data.

Let 0 = Boundary points and A = Interior points.

Count the boundary points and interior points and record in 0 and I respectively.

A 1 _ ()

III

(l)

(3) ....

C

(11)

(4) . . . . (8) . . . (12)

74

1. Fill in the missing numbers from the figure. 2. Find a rule for S (area of a simple closed polygon) using 0 and L.

Boundary Interior Points Point's

Area, S, of simple closed polygon

1 6 1 3

2 9 1 9/2

3 8 1

4 7 2

5 6 2

6 4 4

7 7 7/2

8 . 1 3

i 2

10 6

11 4

12 1 3

Exploring Other Embodiments

Instances of functions abound in nature and in everyday ex-periences. Like the number concept, the fruitfulness of the function concept lies in our awareness of its presence in our efforts to understand and cope with daily technological-cultural demands. The following listing distinguishes be-tween linear and nonlinear functions. In each embodiment, the manipulations (including geometric objects) are dis-played, and partial tables linking the physical or geometric fluents are presented corresponding to entries in the tables.

75

Explore each situation, draw a.graph, and find the rule (function) that is the connecting link between the fluents.

2. Exposed Surface Area vs Number of Rods

Number of rods

rods shifted to 1 cm (Manipulatives)

Linear Functions

1. Blocks vs Height

(Manipulatives)

Number of Blocks

A

0 1 2

Height of Blocks

0

0 2 4

Surface area (in square cm)

A

22 36 50

3. Volume of Rods vs Number of Rods

Number of rods

0

1 2 3 4 5

Volume in cc

A

5 10 15 20 25

76

4. Number of Sides of a Polygon vs Number of Triangles Formed by Diagonals Coming from the Same Vertex

Number Number of

Lz:IIII]of sides triangles

0 A

3 1 4 2 5 3 6 4

(Manipulatives)

5. Number of Sides of a Polygon (Convex) vs the Sum of the Interior Angles of the Polygon

Number Sum of of sides

0

interior angles

A

ii- ------i3 4 5

1800 360° 5400 I . I 6 7200

7 900°

(Manipulative s) Table

6. The Length of a Rectangle vs the Width, with a Constant Perimeter. Let the perimeter = 20 cm

Length width

A 0 -

1 10 0 I 9 J 8 2 7 3 6 4

77

7. Area of Rectangle vs Length (width held constant). Let the width = 4"

Length Area

0

5 20

I 4 16 3 12

I 2 8 1 4

Linear Functions

a. Temperature outdoors vs day b. Height of a plant vs time C. Stock market prices vs day d. Absolute value

0

+3 +3 +2 +2 +1 +1 0 0

-1 +1 -2 +2 -3 +3

e. Area of a square vs the length of its side f. The minimum number of regions formed by chords of a

circle (three chords may not intersect at the same point).

Number of chords Regions

a13**^ 8 10

2 ,_Y T 11

References

1 J.B. Harkin, 'Function Operations and Their Geometric Equivalents,' New York State Mathematics Teachers' Journal, October 1972.

78

New Books Across My Desk

BILL KOKOSKIN

1. Mathematics: Society and Curricula, by H.B. Griffiths and A.G. Howson. Cambridge University Press (Macmillan of Canada), 1974, 423 pages, $10.75 paper and $24.75 cloth.

2. Exploring Mathematics - Insights and Skills, Books I, II, III, by Mon Peters et al. Globe Book Co. (The Book Society of Canada Ltd., Agincourt, Ontario), 1974, for Intermediate and junior secondary students.

3. Algebra 2 with Trigonometry, by Nibbelinz and Graening. Charles E. Merrill Pub. Co., 1975, 566 pages (Merrill Secondary Mathematics Series) for senior secondary.

4.. Consumer Related Mathematics - A Business Approach, by Kravitz and Brant. Holt, Rhinehart and Winston, 1971, 312 pages for junior/senior secondary.

5. Contemporary Mathematics 5 and 6, Elliott, Fryer, Dumbley, Hill. Holt, Rinehart, Winston, Toronto, 1970, 598 pages, and 474 pages each, for junior/senior secondary.

6. Using Advanced Algebra, by Kenneth J. Francis et al. Laidlaw Brothers, (Doubleday), Toronto, 1975, 600 pages, for senior secondary.

7. Calculus - A Programmed Text, Volume I Techniques and Applications, by David M. Merriell. W.A. Benjamin, Don Mills, Ontario, 1974, 460 pages, for senior secondary.

8. Geometry, by Nibbelinz and Graening. Charles E. Merrill Publishing Co. (Merrill Secondary Mathematics Series)

79

Columbus, Ohio, 1975, 532 pages for senior secondary.

9. Algebra, the Language of Mathematics, by Johnson and Johnson, Addison-Wesley, 1975, 600 pages for junior secondary.

10. Mathematics for a Modern World - Book 4, Corli/Eysgard/ Pscia/Del . Grande., Gage Educational Ltd., Toronto, 1975, 500 pages, for senior secondary.

11. Math Probe 2, by Bye and Elliott. Holt, Rinehart, Winston, Toronto, 1973, 282 pages, junior secondary.

12. Motivational Mathematics, by B.G.A. Kelly. Books I to V. Maclean-Hunter (Macmillan of Canada) 1973, about 140 pages each, Grades 7 and .8.

13. Geometry - A Transformation Approach, by Coxford and Usiskin. Laidlaw, (Doubleday), Toronto, 1971, 612 pages, manual available, senior secondary.

14. Contemporary Business Mathematics, by Bello. Saunders, Toronto, 1975, 572 pages, Instructors Manual available, 158 pages, for senior secondary

15. Pre-Calculus Mathematics, by Kal G. Moore. , John Wiley & Sons, Toronto, 1973, 404 pages for senior secondary.

16. Math Is/l, by Frank Ebos. Nelson, 1975, Don Mills, Ontario, 328 pages, for Grades 7 and 8.

17. Mathematics as a Second Language, by Joseph Newmark. Addison-Wesley, 1974, Don Mills, Ontario, 482 pages, junior/ senior secondary.

18. Modern Algebra and Trigonometry, by Vance, '2nd Edition. Addison-Wesley, 1968, Don Mills, Ontario, 432 pages for senior secondary.

19. Using Algebra, by Travers, Dalton, Brunner. Laidlaw (Doubleday), Toronto, 1974, 566 pages, for junior secondary.

20. Algebra I, Doleioni, Wooton, Beckerback. Houghton Mifflin, Toronto, 1974, 556 pages, junior secondary.

80

SPECIAL SSMA PUBLICATION

The School Science and / Mathematics Association

( \ (SSMA) has just published a special 128-page issue dealing with integrated

, science and mathematics courses of study. of the questions that are

/

Some

dealt with in this publi-cation include the follow-

X, % ing: How did single-INC discipline-oriented educa-

tion\ \ p I- r'ri 1

the integration of the disparate content areas

founded on such factors as the nature of the uni-verse, the nature of knowledge and the nature of learners? What are some of the historical events leading to this new 'search for unity'? What are some possibilities for styles and modes of integra -tion of knowledge? Do differing epistemologies (ways of knowing ) among disciplines hinder inte-gration of them? What is the extent and nature of unified courses of study being used in elementary and secondary schools both in the United States and abroad? What have evaluation studies had to say about the efficacy of unified courses of study?

Copies of this publication are available for $1.50 each from the School Science and Mathematics Associa-tion, Indiana University of Pennsylvania, Indiana, Pennsylvania 15701.

81

NEW LESSON AIDS

f

8418 - Primary Mathematical Games and Activities by Prince George Primary Teachers, 11 p. Drawings. 38 ideas. 35.

SEND YOUR 8420 - Geometry Stations for ORDERS TO: Primary by Prince George Primary

Teachers, 7 p. Drawings. BCTF Designed for use by a whole LESSON AIDS class. 35. SERVICE 105-2235 8221 - Busing Around by Hilary BURRARD ST. Strath, 40 p. Drawings. A unit VANCOUVER, designed to guide children to a B.C. V6J 3H9 better understanding of the

transportation system in their community, and to give practice in reading maps, measuring time and handling money. Requires a considerable amount of field work. Primary. Copyrighted by VEEP. $1.

NOTE: PLEASE REMEMBER TO ADD 30% TO THE TOTAL OF YOUR ORDER

PSA75-65/lpW

82

The B.C. Association of Mathematics Teachers publishes Vector (combined newsletter/journal). Membership may be obtained by writing to the B.C. Teachers' Federation, 105 - 2235 Burrard Street, Vancouver, B.C. V6J 3H9. Membership rates for 1974-75 will be: BCTF members $5, BCTF associate members $5, student members (full-time university students only) $1, all other (persons not teaching in B.C. public schools, e.g., publishers, suppliers) $10.

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