BRITISH COLUMBIA ASSOCIATION OF MATHEMATICS TEACHERS

NEWSLETTER/JOURNAL

VOLUME 16, NUMBER 2 NOVEMBER 1974

BCAMT EXECUTIVE 1974-1975

PAST PRESIDENT PRESIDENT J. Michael Baker Alan Taylor 11225 - 87th Avenue 7063 Jubilee Street Delta, B.C. V4C 2Z6 Burnaby, B.C. V5J 4B4 594-8127 (home) 434-6315 (home) 588-1258 (school) 936-7205 (school)

VICE-PRESIDENT Roger Sandford

SECRETARY

R.R. 1, Tzouhalem RoadMrs. Florine Carlson

3043 Spuraway Avenue Duncan, B.C. V9L 1M3 Coquitlam, B.C. V3C 2E4 746-6418 (home) 941-3790 (home) 746-4435 (school) 941-3408 (school)

TREASURER Bill Dale

ELEMENTARY REPRESENTATIVE

1150- 17th StreetMrs. Grace Dilley

2210 Dauphin Place Courtenay, B.C. V9N 1Z7 Burnaby, B.C. V5B 4G9 338-5159 (home) 299-9680 (home) 334-2428 (school) 596-0357 (school)

NCTM REPRESENTATIVE Tom Howitz

IN-SERVICE SPECIALIST

2285 Harrison DriveDennis Hamaguchi

3807 - 22nd Avenue Vancouver, B.C. V5P 2P7 Vernon, B.C. V1T 1H7 325-0692 (home) 542-8698 (home) 228-5203 (UBC) 542-3361 (school)

PUBLICATIONS CHAIRPERSON PRIMARY REPRESENTATIVE Bill Kokoskin Marion Harvey 1341 Appin Road . 946 Gatensbury Street North Vancouver, B.C. V7J 2T4 Coquitlam, B.C. V3J 5J3 988-2653 (home) 939-5488 (home) 988-3161 (school) 939-4201 (school)

NORTHWEST NCTM CONFERENCE CO-ORGANIZERS Trevor Calkins John Epp 3970 Gordon Head Road 1612 Wilmot Place Victoria, B.C. V8N 3X3 Victoria, B.C. V813 5S4 592-4463 (home) 592-2388 (home) 592-1205 (school) 478-5548 (school)

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.

IN THIS ISSUE

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A Note From the Editor ........... Bill Kokoskin

Letter to the Editor

The 3rd Annual Mathematics Summer Printed by John Epp Workshop ............Photos by Bill Kokoskin

No Cold Days in August - The 3rd Annual Mathematics Summer Workshop ........ Walter Szetela

The Textbook as a Creative Tool in Teaching . . . John V. Trivett

Mathematics In-service Questionnaire Results • . Dennis Hamaguchi

Your Executive Committee Bill Kokoskin

A Symposium on the Evaluation of Modern D. Alexander Mathematics Curricula - A Report ........ J. Beamer

W. Higgins

Math Through Activity ............Eileen Maki

An Emerging New Secondary School Mathematics Curriculum ............ Brian Tetlow

Lesson Plans (Pentominoes)

A Problem with An Interesting Result ...... Jay Caturay

Introducing the Metric System ....... James M. Sherrill

Number Theory (A Grade 8 Introductory Lesson) . Jim Ludgate

New Metric Material ............ Bill Kokoskin

New Books Across My Desk .......... Bill Kokoskin

NOMINATIONS TO BCAMT EXECUTIVE

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A NOTE FROM THE EDITOR

Bill Kokoskin

With this issue you should find a catalog from either a publishing company or a company that distributes mathemati-cal laboratory equipment.

Because of rising printing costs and paper costs, we find it is a good idea to subsidize our publication in part.

I am not advocating or trying to sell any particular pro-duct or book to you, my readers, but am trying to distribute

information.

At our next annual general meeting we shall review the policy of sending catalogs, etc., with our Vector. So bear with us, and send your letters of approval or disapproval. about these enclosures, which help to pay for our printing and mailing.

PROFESSIONAL OPPORTUNITIES IN MATHEMATICS

The Mathematical Association of America has recently published the ninth edition of its booklet PROFESSIONAL OPPORTUNITIES IN MATHEMATICS. The pamphlet is designed to give the student and the professional an overview of the job market and salaries for individuals with a mathematical background. The three major categories of employment opportunities for mathematicians are teaching jobs, jobs in government or industry, and jobs in fields related to mathematics, such as computer science and operations research. This brochure also references other professional societies that can provide further employment information for mathematicians.

Copies of this booklet are available for 50 each from the Mathematical Association of America, 1225 Connecticut Avenue, N.W., Washington, D.C. 20036.

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LETTER TO THE EDITOR

Dear Bill:

I am very concerned that the future curriculum revision will have provision for more elective courses at the senior secondary level.

Although there seems to be a lack of a workable, overall, guiding philosophy for the Grade 8 to 10 program, one may consider these years to be exploratory, generalist times.

What options should be open to senior secondary students?

Obviously there should be a good algebra program. There could be two or perhaps three 'semester' length courses. The first course should be an INTRODUCTORY ALGEBRA for those students who have not yet mastered the elementary, algebra skills necessary to cope with more advanced work. For those students who have demonstrated some competence in algebra, the introductory course could be by-passed for the INTERMEDIATE ALGEBRA, which might correspond to the present Math 11. AN ADVANCED ALGEBRA OPTION could be designed upon consideration of the present Math 12 and the following options.

In addition to the Algebra Options there could be options such as:

Mathematics in Trade and Industry Business and Consumer Education Geometry (deductive emphasis) Finite Mathematics (a place of Axioms and

Proofs) Statistics and Probability Calculus Computer Science Humanities Math (similar to Jacob's,

'Mathematics - A Human Endeavor') Trigonometry and Analytic Geometry

Some options would be open to those students without an Intermediate Algebra prerequisite. Some options would be appropriate for the 'general math' student.

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As a closing comment, it seems that schools with elective options have found them to be well accepted.

N. Baker

ANNUAL

HIGH SCHOOL

(c MATHEMATICS

EXAMINATION L 11 MARCH 1975

The Annual High School Mathematics Contest, jointly sponsored by NCTM, MAA, Mu Alpha Theta, the Society of Actuaries and the Casulty Actuarial Society, is scheduled for 11 March 1975.

Registration, which is handled regionally, closes 15 January. A list of regional chairmen can be obtained from Henry M. Cox, 1026 Oldfather Hall (University of Nebraska), 12th and U Streets, Lincoln, Nebraska, 69508. Professor Cox will also furnish order forms for recent examinations and solution keys.

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I'll! -

'SCSs',',G

THE COMMITTEE

3rd ANNUAL MATHEMATICS SUMMER WORKSHOP

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ilk

Pictures taken by Bill Kokoskin, printed by John Epp.

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NO COLD DAYS IN AUGUST The 3rd Mathematics Summer Workshop

Walter Szetela, USC

By almost any measure the third Mathematics Summer Workshop was a clear success. Not only was the registration total a record high, but also comments on the various workshops were exceedingly complementary with respect to quality of the presentations, involvement of the participants, practicality of the activities, and dissemination of ideas. This year's on-site registration was far greater than anticipated, with more than 100 teachers registering on site to bring the registration total to about 375. In addition 41 different people presented workshops so that the number of people involved surpassed 400.

Indicative of the practicality of the workshops and the focus on use of materials was the mass of materials that so many presenters of workshops carted to the various rooms. Boxes and boxes of such materials were carried from cars to rooms as I can personally attest.

During the summer three people committed to presenting work-shops were unable to fulfill their commitments, thus leaving some critical gaps in the program. Fortunately, two people already on the program 'came through' and filled two of the slots. Jim Sherrill, who was presenting a metric workshop for intermediate grades, responded positively and immediate-ly when asked to present a metric workshop for primary grades. Linda Muttitt, who presented an extremely interesting and practical workshop on an individualized station approach in primary mathematics on Tuesday, exhibited her dedication by being willing to present her workshop a second time on Wednesday despite her commitment to a summer job.

Certain workshops should be mentioned on the basis of enthusiastic unsolicited comments I received. Following the keynote address of Prof. Roy Dubisch, Ozan McSweeney presented a metric workshop for primary grades, the people attending being jammed to the walls. Ozan's presentation elicited enthusiastic approval of her activities and ideas and the presider assessed the audience response as 'great' and added the comment, 'terrific ideas for measurement.' Roberta Hay, no stranger to most, delivered two workshops of

two hours each on Math Centers for Learning in Primary Grades, a combination of ideas and activities that left her large audience enthralled, engaged and energized.

Some of the workshops were designed to relate the new intermediate mathematics texts to mathematics teaching. Larry Evans presented practical suggestions for using the new texts along with enrichment activities. He also showed how to interweave school events and activities with instruc-tion on metric measurement. Larry gave this workshop twice to about 75 people. Joy Rufeski also presented workshops on 'Using Math Activity Cards' with Investigating School Mathe-matics textbooks, giving two sessions. For those with experience on Cuisenaire materials, Frank Kennedy and Ruth Kasher presented 'Using New Math Texts With Cuisenaire Materials,' also in two different sessions. Jim Vance presented the 'Teaching of Geometry in Intermediate Grades Using New Mathematics Textbooks' and John Trivett dealt with 'The Textbook as a Creative Tool in the Teaching of Mathe-matics.'

At the secondary level Bill Dale, George Nachtigal, Jerry Bauman and Brian Tetlow delivered workshops on the use of textbooks for the integration of algebra and geometry, on the use of individualized modules for teaching algebra, and on an emerging secondary school curriculum, respectively. John Kiassen and Isabel Leask showed how to use the learning packages they have developed for Grade 8 and 9 mathematics.

Marshal Bye from Calgary, co-author of Project Mathematics and Math Probe, presented three workshops, two for inter-mediate level and one for secondary level, with emphasis on the use of MIRA in teaching geometric principles and con-structions.

One of the most valuable contacts for this workshop was Ralph Gardner, who not only personally presented a very practical and interesting workshop on geoboards in two sessions, but also was instrumental in obtaining the services of no less than six people who presented a total of eight workshops, all of which were valuable additions to the program. The enthusiasm and competence of each of these people reflects glowingly on mathematics programs in Coquitlam.

Most teachers in the Lower Mainland have grown to expect good ideas and a raft of free materials on math games from Tom Flowitz, who delivered workshops on the use of math games

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at both the intermediate and junior secondary levels. Also delivering workshops at two different levels was Dave Robitaille, one on 'Logical Thinking in Mathematics' and one on 'Diagnosis and Remediation in Intermediate Grades.' So many people flocked to attend Dave's latter presentation that a rapid room change had to be effected to seat the vast audience of nearly 80 people.

For the first time in the summer workshops, special work-shops were offered for kindergarten teachers. Responding admirably to this call were Joan Hall, Jean Moutray, Pat Friedrich and Werner Lidtke. Pat Friedrich had no less than ten handouts that allowed participants to avail them-selves of a mass of ingeniously constructed materials rather than take notes. Liedtke also presented a primary workshop on geoboards.

After last year's experience of overcrowding in regular-sized classrooms, some alleviation was afforded this year despite the increased number of participants by presenting a considerable number of workshops twice. Such overcrowding is particularly common in primary workshops as the primary teachers continue to support these workshops through their interest in and openness to new ideas. In addition to work-shops already mentioned, Eileen Maki and Lois Hosein both presented two workshops each on the use of activites, stations, activity cards and laboratory activities. While most of the workshops were one-hour sessions that left no time for pause or reflection, this year a few of the work-shops were two-hour sessions, which gave the speakers more time to introduce and develop their topics before proceeding to group participation. Eva Hilborn, Roberta Hay and John Trivett found the two-hour session more amenable to their workshops.

Some people went above and beyond the call of duty. Grace Dilley, chairperson of site, did a tremendous job, not only arranging for site, assignment of rooms, movement of furniture before and during the workshop, but also pre-senting a workshop designed to motivate mathematics learning in Grades 7 and 8. In addition Grace procured the services of Sue Mikitka, who prepared the multitude of signs that assured a smooth flow of traffic to the right rooms - a job that must have taken untold hours. It is jobs like these, innumerable but necessary, that usually go unnoticed but are vital to the success of a workshop.

Having been chairperson for the Math Workshop two years

running, I know full well the multitude of tasks such an undertaking involves and therefore the value of a good working committee. All of the committee members contributed much in time and effort through numerous meetings and completion of job assignments. Yet, the work of some committee members was so meritorious and unflagging that these people deserve special mention. Hugh Elwood, the lone holdover from last year's committee, contributed much toward effectively publicizing the workshop despite the committee's late start in initial organization. Ken Silen, treasurer, looked after numerous tasks and was always ready to do what-ever was asked. There are few who are as dependable, reliable and unfailing as Ken. Shirley McGill processed hundreds of registrations, and stuffed packets with Workshop information and free-materials. Grace Dilley devoted hours and days to arranging for site and taking advantage of the accommodations at Queen Elizabeth Secondary School in Surrey.

A word needs to be said also about the co-operation of school administrators at Queen Elizabeth. Every request and inquiry was given complete attention by Principal Sam McDowell, and Mike Baker, vice-principal expedited arrange-ments for use of school equipment and facilities.

Despite the success and satisfactions of the third Summer Mathematics Workshops, some suggestions for the future emerge out of problems encountered this year. The following is a list of recommendations for the chairman and committee for next year's workshop.

1. Start early! There is too little time to arrange the multitude of workshops, presiders, procedures, publicity, etc., by starting in February with a program expected to be in the schools in June.

2. Take greater advantage of those people qualified to present workshops. A person who repeats a workshop gives more flexibility for participants to avail themselves of useful workshops while helping to keep down excessive numbers for any one workshop. Classrooms are

• not designed to accommodate 80 partici-pants adequately.

3. Use more of the site's facilities instead

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of limiting yourself to a minimum number of rooms. In this way those who present the workshops can transport their materials to rooms in plenty of time instead of having to wait for another person to complete his workshop in a certain room, thus having only fifteen minutes to clear the room of old materials and make room for new.

4. Allow for cancellations by persons committed to a workshop. It may be wise to plan extra workshops, especially at primary sessions.

S. Do something about encouraging more senior secondary school teachers to attend. The problem is not easily solved. Knowing that secondary school teachers do not attend in large numbers, a much smaller part of the program is devoted to secondary school teachers. Because there is less offered for secondary school teachers, fewer such teachers are induced to come.

6. Anticipate the weather. It can get hot in August, with 50 bodies in a room and no air-conditioning.

7. This year a number of student teacher graduates were attracted to the Workshop at a half'rate. This policy should be continued and publicized at the universities.

The Summer Workshop was invaluable for the teachers in attendance. Perhaps more significant is the number of extremely capable people who presented workshops and are now known to the BCAMT. These people can continue to render great service to many more teachers throughout the province. Dennis Hamaguchi, In-service chairman, will no doubt be pleased to enrich his file of talented workshop personnel with many of the people 'discovered' in this year's Summer Workshop.

As outgoing chairman of the Mathematics Summer Workshop for the second year, I wish to thank all of those who partici-pated or assisted in some way. Special thanks go to Marian McDonald who helped the committee get under way and proposed

an August Workshop despite the reservations held by some, which turned out to be so well attended. May next year's chairman be as fortunate in having so many people make such a concerted effort.

ERIC PUBLICATIONS ON EVALUATION

Two publications on evaluation written by Marilyn N. Suydam have been developed at the ER IC Center for Science, Mathematics, and Environmental Education (ER IC/SM EAC).

UNPUBLISHED INSTRUMENTS FOR EVALUATION IN MATHEMATICS EDUCATION: AN ANNOTATED LISTING (264 pages, $2.75) gives specific information about over 200 investigator-developed tests and other instruments to assess mathematical instruction. EVALUATION IN THE MATHEMATICS CLASSROOM: FROM WHAT AND WHY TO HOW AND WHERE (70 pages, $1.75) discusses the role and the scope of evaluation in the mathematics classroom, with specific suggestions on objectives, purposes and various types of evaluation pro-cedures.

These publications can be ordered from the Center for Science and Mathematics Education, 244 Arps Hall, The Ohio State University, Columbus, Ohio. 43210.

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THE TEXTBOOK AS A CREATIVE TOOL IN TEACHING MATHEMATICS

John V. Trivett, SFU

The situation studied in this session of the BCAMT Third Mathematics Summer Workshop in August was a classroom set of those much-maligned arithmetic textbooks, with the challenge to come up with creative uses probably not intended by the authors! Intermediate texts were available on this occasion, so all the suggestions made by participants working in groups are not necessarily applicable to every set of books.

Before all shared their creative skill: two examples were given, one as an aid to teaching fractions, the other about symmetry.

The group first agreed that although in fact every book is different from every other book, the nature of sets of texts is such that it is easy to assume that every book is the same as every other. That is to say, any one can be sub-stituted for any other and this is especially true when the books are new and no graffiti, names or thumb prints add to the differences!

It was agreed, moreover, for the purposes of the study, that whatever attribute was assumed about one book was precisely the same for every other book. Every attribute and every value of every attribute - ordinary or extraordinary!

Two subsets of the books were piled up for all to see and the question was posed: 'What two numbers give the relation-ship of this subset to that subset?'

A participant quickly replied, 'Six, five.'

The leader, registering slight surprise, said, 'Not twenty-four, twenty? What did you count?'

'The books.'

'Oh! of course. I was thinking of the words on the covers!' On the blackboard therefore were symbolized the two equiva-lent pairs (24,20), (6,5), they both representing the relationship.

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Others began to offer answers:

(48, 40)... 'I've included the words on the bindings.' (12, 10)... 'That's the covers, two to each book.' (72, 60)... 'There are twelve chapters in a book.'

These were put on the board along with (6 x 12, 5 x 12) as another form of the last entry.

(15, 18)... 'There are three children pictured on the cover of each book.'

This one was immediately challenged, someone asking, 'Hasn't the order been changed? Does it matter?' The leader said, 'Well, let's write (15, 18) over the other board so we can add members of either family, one that has the smaller number first, the other the larger number first.'

Gradually, attributes and their values became more imagina-tive, even wild, and no-one really cared whether they were attributes relevant to the actual books. One teacher said that he hadn't the faintest idea how many ink marks there were in a book, but if there were 2,485,672 marks in one book, a pair belonging to the (24, 20) family was (2,485,672 x 6, 2,485,672 x 5). Another pair given was (184 x 6, 184 x 5), thinking there might be 184 letter x's in every book. Very soon everybody was generating many members of the family because the patterns for such a generating process were seen and the actual books could be disregarded.

Summarizing, it was said that this could be the beginning of a valuable procedure in the early study of rational numbers as sets of ordered pairs of whole numbers. The rational called 'six fifths' is the class of pairs [(6, 5), (12, 10), (18, 15)... (6n, Sn)....)]. (5, 6) is the reciprocal class.

That concluded, everyone was asked to open a book flat on his hand.

'Consider the crack as a line of symmetry. Is there any mark on a lefthand page that will coincide with the 'same' mark on the opposite page when the book is closed?'

(The beauty of this is that the result is never seen, as at the moment of impact the marks are out of sight. One is forced therefore to use imagination and inferences about

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symmetry!)

Examples found included:

• Red rectangles used for the backing of rubrics. • Underlinings of multiplication examples. • Period dots. • The rectangle on each page inside of which all

symbols have to be printed. • Occasionally throughout the book, parts of

drawings. • The 0 of page number 106 and 0 of 107, but not

the l's. (What other digits?) • Occasionally a plus or minus sign.

There are undoubtedly many more, once one really looks.

The group work produced the following suggestions - obviously they are not intended to be very profound or sig-nificant. Take your pick or disregard the lot. But please use a textbook creatively!

Geometrical

What geometrical figures can we find' in the book or attrib-ute to the book itself?

How many horizontal, vertical or parallel lines are on one particular page?

If there is a map in the book with places marked, how many ways can you go from one place A to a place B?

Can you trace a path from A, through B, through C, etc., without retracing or crossing paths and yet visit every place?

Arithmetical

Where there are sets of 'problems':

8 6 7 7

x8 _x8 ... j x8 and soon,

ask also 'What is 8 x 8 x 6 x 8 x 7 x 8...?', as far as the children like to accept the challenge.

Ask also 'Which answers will be even, which odd?'

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Ask also 'Can you arrange these products in order of size, without multiplying and with multiplying?'

'How many ways can the pages numbers of a chapter be added?'

Area

Use the book as a unit of area. How many would cover the floor?

'How many 1 cm by 1 cm square pieces of paper would cover the cover?'

'What size sheet of wrapping paper is best to wrap the book for sending by mail? What length of string would be needed? (Allow for a knot.)'

Estimation

Where is the center of the book?

What is its length, width, thickness, mass?

General

Use the book as an obstacle in PE for jumping over (Shades of Evel Knievel!).

Write a new title for the book.

Rewrite one of the pages so that someone your own age would understand it better.

Redesign the cover.

Dramatize how the book was made.

Dramatize what it is like to be a small number.

Read a page upside down and write the working of a problem upside down too! (Be careful of the 6's and 91s.)

Compare parts of this text with another old book.

Lastly, one suggestion considered very important and worthy really of much more time than the session allowed: Instill into students that exercises set in textbooks are always invitations to invent others of your own and that the ones

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you create for yourself are more important than those printed. One has therefore never finished, more challenges

remain.

BCTF RESOURCES CENTER

Recently acquired material available for loan. Hours: Monday-Friday, 9-5; Saturday, 9-1. Phone: 731-8121

James, Roy G. Metrication for administrators. Toronto, Ontario Teachers' Federation (1973). 43 p. QC91/J36

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MATHEMATICS IN-SERVICE QUESTIONNAIRE RESULTS

Dennis Hamaguchi

Last March, a mathematics in-service questionnaire, which was jointly sponsored by the BCAMT, the Math Education department of UBC, the Extension department of IJBC and the BCTF, was sent to every school in the province. The results of the questionnaires have now been compiled, thus com-pleting the first phase. It is my hope that in-service workshops can commence this school year.

We received completed questionnaires from approximately 490 schools, with approximately 4,000 teachers participating to complete the questionnaires. The following is a summary of the kind of school that completed the questionnaire:

Elementary 376 schools Junior Secondary 50 schools Senior Secondary 35 schools Junior-Senior 23 schools Elementary-Junior 3 schools Elementary-Senior 4 schools

The questionnaire results seemed to indicate a rather con-sistent pattern throughout the province. Regardless of the geographic area, the teachers indicated a strong preference for the same workshop topics in their respective grade levels. The following is a list of suggested workshop topics that were received very favorably.

Grades 1, 2, 3.

1. The Use of Stations 2. Metric Measurement 3. Geometry in the Primary Grades 4. The Use of Physical Aids

Grades 4, 5, 6.

1. Development of Problem Solving Skills 2. Effective Uses for Games and Activities 3. Diagnosis and Remediation 4. Laboratory Methods for Intermediate Grades

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Grades 7, 8

1. Teaching in a Multi-text Environment 2. Problem Solving 3. Maintaining and Updating Computational Skills 4. Diagnosis and Remediation

Grades 9, 10

1. Activities in Algebra 2. Activities in Geometry 3. New Topics in the Secondary Curriculum

Grades 11, 12

1. Activities in Algebra 2. Topics and Methods for General Mathematics 3. New Topics in the Secondary Curriculum

Geographically, the questionnaire results indicated the obvious - highly populated areas showed the heaviest returns. Primarily because of financial limitations, it would not be feasible or practical to hold workshop sessions in every city. It is most likely that the province will be divided into geographic areas as indicated roughly on the chart.

In the coming months contact will be made with in-service chairpeople, superintendents, district staffs to implement pilot workshops. Tentatively, the procedure for imple-menting workshops will be as follows:

1. BCAMT in-service specialist will ask school districts about the possibility of a work-shop as indicated by the questionnaire results.

2. If the district(s) concerned wish to have a workshop, they can supply such pertinent information as place, time, date and length of workshop.

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GEOGRAPHIC GROUPING OF SCHOOLS FOR MATH IN-SERVICE QUESTIONNAIRE

Area and Districts Elem. Jr. Sr. J-S E-J E-S

East Kootenay 19 3 1 1 2 (1, 2, 3, 4, 7, 86)

West Kootenay 9 2 3 (9, 10, 11)

South Okanagan 12 2 1 (14, 15, 16, 17, 23)

North Okanagan 14 4 1 (19, 21, 22, 89)

Kamloops Area 23 1 4 (24, 26, 30, 31)

Cariboo Area 18 1 2 (27, 28)

Upper Fraser Valley 27 6 2 2 (33, 34, 35, 75, 76)

Lower Fraser Valley 50 10 6 (36, 37, 42, 43)

Vancouver Area 78 5 4 9 (38, 39, 40, 41)

North Shore/S. Coast 25 2 3 (44, 45, 46, 47, 48, 49)

Prince Rupert Area 12 1 2 1 2 (50, 52, 54, 80, 87, 88)

Prince George Area 16 2 (56, 57)

Peace River Area 10 2 1 (59, 60, 81)

Victoria Area 33 2 6 1 (61, 62, 63, 64, 65, 66)

Nanaimo/Courtenay Area 30 9 4 2 (68, 69, 70, 71, 72)

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YOUR EXECUTIVE COMMITTEE

Bill Kokoskin

This is the second in the series of brief activities featuring your executive committee. This time I'm featuring myself, the Publications Chairman.

I'm in the second year of a two-year term as publications chairman for the BCAMT.

I received all my high school training in Montreal, and received my B.Sc. degree, with a major in mathematics, from the McGill University. After migrating west, I completed my one-year teacher training at UBC and then completed my M.A. in mathematics education, also at UBC.

My teaching experience includes four years at Argyle Secon-dary School, where I taught all levels of mathematics from Grade 8 to 12. I am now department head at Hamilton Junior Secondary School in North Vancouver School District.

In October I was appointed to the Department of Education's Mathematics Revision Committee.

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A SYMPOSIUM ON THE EVALUATION OF MODERN MATHEMATICS CURRICULA - A REPORT

D. Alexander, University of Toronto J. Beamer, University of Saskatchewan W. Higgins, Queen's University

At the International Congress of Mathematicians, Vancouver, August 21-29, 1974 a three-day symposium was organized by the JCMI (International Commission of Mathematical Instruc-tion) to discuss evaluation of modern mathematics curricula. Reports were presented by representatives of the United Kingdom, Russia, Poland, Brazil, Japan, India, U.S.A., Canada, Germany, Denmark and Hungary, with a general discussion on the topic on the third day. An official report will be sent in due course to all affiliated organi-zations (in Canada, the Canadian:Mathematical Congress). This present report reflects the impressions of the Canadian representatives.

There appeared to be general agreement that initial evalua-tions conducted in the experimental phases of the new curricula had been favorable, while the evaluations became less significant as the curricula were expanded to encompass more schools. Reference was made to three factors that could explain this: (1) the reduction in the selectivity of the secondary and tertiary student bodies which coincided with the introduction of the new programs in most countries; (2) the lack of adequate teacher training for implementing the new. programs; (3) the inability to define 'modern mathematics curricula' (is it primarily content or method and if so what content and what method?).

The only large-scale study reported was the National Long-itudinal Study of the U.S.A. where the most important factor determined was the teacher. No characteristics were identified for the effective teacher and the effectiveness of the teacher was not consistent in successive years. It is expected that the National Assessment of Educational Progress will contain some evaluation of mathematics curricula in the U.S.A.

undertaken that will be of statistical. There is a • statistical studies are many uncontrollable

In the U.K. a study is just being an observational type rather than definite feeling that large-scale doomed to failure because of the variables involved.

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Dr. Christiansen (Denmark and UNESCO) emphasized the need for evaluation of curricula on the basis of well-established goals (i.e., the goals of a developing country like Brazil, where illiteracy is a major problem, are entirely different from those of the U.S.A.). He also referred to his experience with UNESCO as impressing him with the impossi-bility of successfully 'transplanting' curricula. 'Each country must find its own salvation.'

The Polish answer to teacher training was enlightening. At 4 p.m. on a given day each week every elementary teacher will be required to watch an in-service TV program and assignments will be marked weekly by the local superinten-dent. This is one way of solving the teacher training prob-lem. (It was not clear what happens to delinquents.)

Although no conclusions were reached on the evaluation of the modern mathematics curricula or on how such evaluation should take place, nor indeed even whether it should take place, the symposium did provide a forum for an exchange of information and opinions of an international level and was valuable for this alone.

The ICMI is a Commission of the International Mathematical Union. Canadian representation is through the Canadian Mathematical Congress. Dr. A.J. Coleman, President of the CMC, hopes to institute a mechanism for making this repre-sentation more effective in the future through the forma-tion of a national committee or commission for mathematics education. It is hoped that he will find the support, political and financial, for such a committee.

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MATH THROUGH ACTIVITY

Eileen Maki, Begbie Elementary School

Purpose - To establish a 'gut-level' feeling for concepts fundamental to mathematics.

The Use of Manipulative Materials

It is important to note that manipulative materials are not toys or 'baby-stuff.' They are mathematical tools and should be introduced to the student as such.

I suggest that the introduction should consist of a brief explanation of the tool's background, if this is available, and of an 'open' problem for the students to explore. This should lead into the first phase - 'messing around.'

Phasc One

1. Includes the introduction of the tool. 2. Students explore both teacher-directed and student-

directed questions. 3. Discussion between teacher-pupil and pupil-pupil about

their findings is necessary. 4. If the 'messing-around' is done seriously, the student

will leave this phase with a 'feeling' for the topics he has explored and for the materials he has used. This feeling will include a basic knowledge of the tools used and a curiosity about what else is possible.

Phase Two

1. This is a period of intensity. The 'feelings' of Phase One are solidified and extended.

2. Specific topics are explored in depth or a specific tool is explored to its fullest possible extent (that is, for the child involved and the goals set).

3. Various findings from Phase One are studied. For example, Why did John get this answer and Sue another answer? What operations were performed? Does John's method always 'work'? Does Sue's?

4. At this stage, 'correct' answers should make sense to the student because he can understand and see what was done to 'get' the answer. Children will know why some answers are 'better' than others. It follows that they will also know why some methods are better than others.

22

Consistency of a method in providing acceptable results will be appreciated.

S. When to use an approximation rather than an exact number will be recognized. Students will acquire the skill of guessing and will know when this can be a valuable tool.

6. Mastery is stressed. 7. This period should include both station-type activities

and large group or class activities. This is advisable so that the strengths of both approaches can be drawn on.

For example, when a concept is first introduced, one can usually count on two or three students to 'get the ball rolling.' A large group is necessary to take advantage of this. However, once the idea is taken up by a few more students, it is often advisable to group. This encourages more students to become involved.

Phase Three

1. This is a period of extended exploration.

2. Those students who have finished Phase Two with concept mastery will proceed to project work. They will collect data, evaluate them, and form conclusions about their problem. The projects they undertake may be teacher-, other student— or self-directed.

3. Those who have not finished Phase Two at a satisfactory level will receive more guided instruction. The prob-lems they pursue will be geared to help them with a specific area in which they need extended 'practice.'

4. The problems may be worked on individually or in small groups.

Introductory Period - HOW TO START

A. Characteristics

1. Students are on Phase One of manipulative materials. They are familiarizing themselves with the materials they will be using.

2. The teacher is able to work with small groups to establish the students' individual levels.

3. The teacher spends the bulk of her time with the instruction group. However, some time is spent with each of the other groups each period.

23

B. Example of Stations - 6 PERIOD WEEK

Geoboards

Tangrams Math Kit

Instruction Group Rods

Setsp lay Instruction Group

Stencils Attribute Pieces

Math Games Stencils

Cubes, Squares and Rods

1. Examples have been given of two different cycles. Each cycle might last two or three weeks.

2. Note how more active stations are alternated with less physically active stations. Mental activity remains high throughout.

3. Instruction Group - This gives the teacher and students an opportunity to explore together. The instruction may be the introduction of a new tool or a new concept. It may serve diagnostic or review purposes.

C. Grouping

The stations are most successful when the grouping is heterogeneous.

Strengths of grouping:

1. each student feels group pressure to concentrate. 2. responsibility must be accepted by students for their

behavior. 3. discussion is less inhibited.

Example of a Specific Study Topic

Topic - AREA

Phase One - This will extend over a very short period of time. The students are already familiar with all of the equipment that will be used. A discussion of what area' means to the students is necessary. This would probably be done in a large group. A question might then be asked and explored. For example: Which has more area, this table or that door? (If the objects chosen are similar in area, the discussion is more involved.) The discussion here should lead into ways of studying area more intensively.

Phase Two - This will include the use of many materials and

24

much discussion. Examples of materials that might be used are:

• Geoboards • Tangrarns • Cubes, Squares, Rods • Stencils • Activity Cards • Plenty of paper, string, boxes of various sizes,

measuring devices.

Phase Three - Students do extended exploration. This in-volves Research Activity Cards and Experiment Cards.

Topic - VOLUME - CAPACITY

Phase One - The 'messing around' stage of this topic could be introduced by asking the students which container, of a number set before them, would hold the most sand. Children could discuss this, come to their own conclusions and then test these.

• Containers varying in size and shape should be available.

• There should be a variety of filling materials available - water, sand, paper, rice, marbles. Children can explore whether the container that holds the most sand is the one that holds the most of every other filling material available.

• Permanently filled materials, such as blocks, balls, closed jars, should be available. Which is the biggest, smallest could be explored?

Phase Two - Children will be guided into exploring and testing some of their ideas and theories obtained from Phase One in a more intensive manner. Students will be questioned to show and explain why they believe that some-thing is so.

Teacher-prepared activities will be available and students will be expected to involve themselves in most of these. Examples of materials that will be used are:

• Cubes, Squares, Rods • Stencils • Containers of various sizes • Filling material • Geoboards and straws

25

• Graph paper • String

Phase Three - Students use Research Cards and Experiment Cards to extend their exploration of the topic. Any ques-tions they wish to explore further are researched here.

Area and Volume

Area and volume can be related in many ways. Students may wish to explore how the outside area of an object relates to its capacity or mass.

Students may make charts Or graphs to show relationships and to determine if they feel they can make a 'rule.'

Extra Notes

1. Some students will not need to use all of the materials before going on to Phase Three.

2. Some students may need more materials to master Phase Two. Some may need more time at one station than at

• others. 3. Students should always be encouraged to do their own

examples of an activity. They should keep records of what they have done. Example:

Date Activity Prepared Work My Own Work

April 19 Geoboards 2 cards 3 questions

4. A good idea for marking is to have students indicate those questions they know they have correct, those they are uncertain about, and those they didn't understand.

Example: A check beside a question number means they are certain about it.

• A question mark indicates uncertainty. • A circle around the question number indicates that they

had problems with it for one reason or another.

26

AN EMERGING NEW SECONDARY SCHOOL MATHEMATICS CURRICULUM

(with reference to the textbooks and suggestions for implementation)

Brian Tetlow

The question is often asked, 'Why is it necessary to change our mathematics curriculum so frequently?' A revision has traditionally taken place at least once every ten years and I suppose that it is only right to try out new ideas and to discard that part of the curriculum that seemed inappropri-ate. Usually 'revision' means a change of textbook; the courses have changed little over recent years.

With a change in emphasis in elementary education, students have been allowed to enter Grade 8 with widely varying mathematical backgrounds and this has caused some problems for secondary schools. The future secondary school mathe-matics curriculum has been designed to cater to student differences, to provide a broad mathematical education for all and to be as flexible as possible.

The Mathematics Revision Committee was faced with some tough problems when it started its work on the revision of the junior secondary school mathematics curriculum. It was felt that General (nonacademic) math courses were too restrictive and should therefore be replaced by broader courses so that the door to an academic mathematics program in the senior secondary grades would remain open to more students. If all students are exposed to the same core program up to year 10, it would be possible for any student to select any elective math course in Grade 11.

At this point you may be jumping up and down in frustration and yelling, 'How can all students study the same mathemat-ics? That's impossible!' Well, let us examine the CORE program for years 9 and 10 as recommended by the Revision Committee. A revised curriculum guide including this preliminary draft may be available to all schools of the province in September 1974. It is anticipated that these courses will become available for permissive implementation in September 1975. It is hoped that supporting texts will be made available by the spring of 1975 so that schools wishing to offer the new courses can decide which texts to order.

27

I plan to give only the main highlights of the curriculum since full details will be available soon. What are the main differences between the present Grade 9 and 10 courses and the proposed courses?

First, each course will contain some algebra, some geometry and some consumer mathematics. The topics in algebra. are largely the same as those in the present Grade 9 (academic) course with the exception of proof, which is omitted. The deductive geometry that is now part of the Grade 10 course has been replaced with intuitive geometry. The main headings for geometry are as follows:

A. Parallel lines B. Pythagorean Theorem C. The Circle D. Similar figures (trig-Grade 10) E. Symmetry F. Measurement (perimeter ) area, volume and angle calcula-

tions) G. Vectors

In addition to algebra and geometry, each course contains some consumer mathematics, an area that has been sadly neglected in math courses, particularly for academic stu-dents. It is hoped that all students will benefit from .a study of simple and compound interest. In addition, topics in probability, statistics and the computer (the latter is optional) are included.

It will be obvious to the reader that the new program differs appreciably from the present program. It seems that many of the topics are academic in nature and would there-fore be unsuitable for slower and less able (nonacademic) students. The method of teaching these topics presents a problem to the teacher and at this point I should like to make a few suggestions for your consideration.

I think it would be unreasonable to expect every student to study each topic to the same depth. An able student will be able to study a topic in great depth and in a far more sophisticated way and possibly faster than could a less able student. For example, let us consider the topic 'simple inequalities.'

28

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Students of below average ability

Students with average ability

Students with above average ability

examples like examples like examples like

2x < 6 3x + 1 > x + 7 5(3x - 1)< 17x - 9 5 >2x 4(x+2)'z2x+1 2 5 2x + 1 > 7

- 3)>--x + 4

As a second example, consider the year 10 core topic 'Quad-ratic equations by using the formula.' You may feel that this topic is too difficult for your students. Fine - then leave it out of your course for those students who would not benefit from exposure to this topic. Remember, the Curric-ulum Guide is purely a guide and you are not duty bound to teach everything to every student. You may, in this parti-cular case, decide to accept solutions by trial. For example, a slower student should be able to solve: x 2 - 3x + 2 = 0 by using the trial method.

)O many topics to be are is considerable over-e at the school or dis-topics until the Grade 10 time available during the could be selected from 'level 1' and 'level 2' few pet topics of your

At first glance there seem to be t covered in each year. However, th lap and it may be necessary to agr trict level to leave some Grade 9 year. Perhaps there will be some year for enrichment topics. These the more difficult topics labeled in the guide, or perhaps you have own.

The diagram below summarizes my suggestions so far.

Group Percent K 1 YEAR

Above Av 20% Core(high level of sophistication) En Average 40% Core(Av level of sophistication) En Below Av 40% Core(low level of sophistication) En

En = Enrichment Av = Average

The next problem facing you, the teacher, is the problem of which texts to use with a particular group of students. The scale of issue will be 150% of enrollment, which allows the teacher some degree of flexibility. Although the titles of texts have not, to my knowledge, been announced, you will be able to choose from at least three text series. I very much

37

doubt whether one text will adequately cover all topics, so the problem remains - how to switch to another text for more than half of the class. You have no doubt solved this prob-lem already in your mind. However, one idea that may be useful to you is the following one:

Suppose you teach two Grade 9 classes of 30 students.

Total students = 60 : Total texts allowed = 90 (= 3 sets)

If you have room, you could store the three different sets of texts in your classroom. Texts would be issued and collected for each class. The home-study problem could be overcome by a sign-out system. Homework assignments could be copied into notebooks or issued on Ditto. The texts would normally be numbered on the outside sothat students are responsible for a text and only use that text.

To conclude, I should like to mention one text series I have had the opportunity of piloting over the past year. Mathematics for a Modern World (Gage) was a popular series with the students. They particularly liked the provision of the answers to alternate questions on each page in the mar-gin. Students found it fairly easy to work independently with the texts - plenty of worked examples are provided. I believe that this text is being seriously considered by the committee. Books 1 and 2 contain most of the core topics at an average level of sophistication. For the less able students Essentials of Mathematics (Ginn) Books 3 and 4 seemed very appropriate for much of the core at a low level of sophistication. It will be a pity if these texts are not on the list of recommended texts.

Let us not be over-critical of the Revision Committee's recommendations. The changes they have suggested are indeed major and the committee has been courageous in making them. With careful thought and planning by the teacher the new curriculum can then be successfully taught. Homogeneous grouping may be desirable to maximize efficiency, but it is not necessary. For less able students it may be desirable to allow more time to cover the courses.

You may have wondered what the plans are for Grades 11 and 12. I believe that the Grade 10 graduate will have several Grade 11 math courses to choose from, including a course on deductive geometry. If there is a fairly wide choice at the Grade 11 level, I foresee few problems. And, theoretically, the door has remained open for any Grade 10 graduate to select any Grade 11 course. I hope the students make wise choices

38

A PROBLEM WITH AN INTERESTING RESULT

Jay Caturay

Problem: Find the maximum number of regions R formed if several points are chosen on a circle and connected with straight lines, as a function of the number of points (n).

For example:

Given two points on the circle and then connected with straight lines, we can form two regions, as in Figure 1.

Figure 1 Figure 2

Given three points on the circle and then connected with straight lines, we can form four regions, as in Figure 2.

Solution:

Working out the problem for the first eight points, I found the following results:

number of points (n) 0 1 2 3 1 4 1 51 61 7 regions formed (R) 1 1 2 4 1 8 1 16 1 31 1 57

Note that for every given value of n (in an arithmetical progression), the value of R assumes a series of values. Hence the calculus of finite differences can be applied in this situation.

The calculus of finite differences is not a very popular branch of mathematics except to those interested in recre-ational mathematics. The calculus of finite differences had its origin in Methodus Incrementorum, a treatise published by the English mathematician Brook Taylor (of Taylor's Theorem).

To apply Isaac Newton's remarkable formula for solving

39

problems where we can apply the calculus of finite differences, we make a chart as before, then forming rows, each representing the differences of adjacent terms in the row above until the numbers in the row are all alike:

number of points 0 1 2 3 4 5 6 7

regions formed 1 1 2 4 8 16 31 57

1st Difference 0 1 2 4 8 15 26

2nd Difference 1 1 2 4 7 11

3rd Difference 0 1 2 3 4

4th Difference 1 1 1 1

The number of rows of differences where the numbers are all alike is the order of formula. In our case, the order is the fourth. Newton's formula for the nth number of the series is:

A + Bn + Cn(n - 1) + Dn(n - 1)(n - 2) + 2! 3!

En (n - 1)(n - 2)(n - 3) + 4!

where A is the first number of the first row of difference where B is the first number of the next row of difference and so on. In our problem, we have A = 1, B = 0, C = 1, D = 0, E = 1.

Hence, the solution of the problem, the number of regions (R) formed as a function of the number of points (n) is:

R = 1 + n(n - 1) + n(n - 1)(n - 2)(n - 3) 2! 4!

After getting this expression, I noticed that I can replace

n(n - l) by (n 2!

and n(n - l)(n - 2)(n - 3) by (--) 4!

40

the solution to the problem becomes:

R =. 1 +(n'1 + ( n 2)

but In\ In - 1'\ In - Ii) 2 ) ( i )

and

fn\ (-!!= 4 ) + ( 3

Rewriting the expression for R we get:

R= 1 + fn - fl +(n - 1+fn - 1+fn -1 k 1 1 1. 2 ) 3 ) 4

As soon as I got this expression, Pascal's triangle came to my mind. The surprising result is that the number of regions (R) formed if several points are chosen on a circle and then connected with straight lines is none other but the sum of the horizontal numbers on the left of the line on the Pascal's triangle (except that the n on the expression for R is equivalent to n - 1 on the Pascal's Triangle).

n=0 1 n 1 1 n=2 1 2 1 n=3 1 3 3 1 n=4 1 4 6 4 n=5 1 5 10 10 n=6 1 6. 15 20 l/'6 n=7 1 7 21 35 35V'21 7 n = 8 1 8 28 56 70V' 56 28 8 n = 9 1 9 36 84 126y'126 84 36 9 n = 10 1 10 45 120 210 252 210 120 45 10

RI

161 311 57 99

163 256 386

41

INTRODUCING THE METRIC SYSTEM

James M. Sherrill, University of British Columbia

Canada is committed to the metric system of measurement. You may be aware of this but may not have a clear idea of exactly what the metric decision means to you as a teacher. It is hoped that this section will serve three purposes:

1. give you an idea of how the metric decision will affect you,

2. help you understand the metric system of measurement, and

3. give you some hints for teaching the metric system of measurement to your students.

History and Rationale

The English system of measurement developed from man's need to measure size and distances using units from the most readily available object - himself. He utilized his palm, span, finger, an ell, and a fathom for length; his foot, step, pace, an arrow's flight, and a day's journey for distance; and a handful, sheilful, hornful, or gourdful for capacity.

There was little need for standardization until man began to travel and trade with other men. When 'standard units' were developed, a new problem arose. Different countries used different definitions for the same unit. The foot was, at first, the length of any man's foot. In some countries, it was the length of the king's foot (since he was the 'ruler') and this foot could change as the 'rulers' changed. Later an effort was made to standardize some units; for example, England and Scotland decreed the foot to be 12 inches. Unfortunately, England and Scotland didn't use the same definition for the inch.

Today; in the age of technology, one still finds different units in those countries which are not yet metric. Canada and the United States are neighboring countries, yet they use two different definitions for the gallon. A question at which people in metric countries must laugh - 'Which is heavier, a pound of gold or a pound of feathers?' A pound of feathers is heavier since feathers are weighed by the avoirdupois pound (1 avoirdupois pound - 7000 grains) and gold is weighed by the troy pound (1 troy pound - 5760

42

grains). Which is heavier, an ounce of gold or an ounce of feathers? An ounce of gold is heavier. There are 12 ounces in the troy pound, so one ounce of gold weighs 480 grains; there are 16 ounces in the avoirdupois pound, so an ounce of feathers weighs 437.5 grains.

,_-3) ^'

Out of such confusion there developed a need for a simple, standardized system of measurement. In 1670 Gabriel Mouton, a French abb6, developed a system of measurement organized according to the decimal system of numeration. It took over a hundred years for a system of measurement like the one Mouton put forth to get official sanction. In 1790 the French National Assembly appointed a committee to study the measurement situation and see if a rational system of measurement was possible. In 1795 France adopted a decimal system of measurement, defining the base unit of length to be the metre (from the Greek word metron a 'measure').

The metric system did not use parts of the human body as units. The metric system did not develop haphazardly adding more and more units as the need arose. The metre was defined as one ten-millionth of the distance from the North Pole to the equator, along the meridian passing near Dunkirk, Paris and Barcelona. One can see that such a definition would be difficult to replicate in any one country. Also, the length of the metre changes as the position of the North Pole changes; at the time that the metre was defined, scientists were unaware that the position of the North Pole changed.

43

In 1870, because of the problem of replicating and comparing metric units from country to country, France called a meeting of the metric countries to develop a 'unified metric system of measurement.' In 1875, the Treaty of the Metre was signed to establish the General Conference on Weights and Measures, which meets to determine the official defini-tions for the units used in the metric countries. In 1960 the Conference adopted the Systeme International des Unites (SI). It is this SI metric system that is most used throughout the world.

A Popular System

The popularity of the metric system stems from two charac-teristics - the high degree of standardization and its simplicity.

In the entire metric system there are only seven base units! They are metre (length), kilogram (mass), second (time), ampere (electric current), degree kelvin (thermodynamic temperature), candela (luminous intensity), and mole (amount of substance).

All units used in the metric system are related to these seven base units. The units you will be most concerned with (because they are the ones used in everyday living) appear in Table 1:

Table 1: Metric Units to be Studied

Quantity Unit Symbol

Length metre m Mass kilogram kg Capacity litre *

Temperature degree Celsius °C

* As a rule of thumb, the cursive letter (9) is used as a symbol for the litre to avoid confusion with the numeral (1): however, in symbols such as ml (millilitre), ki (kilolitre), the cursive form is not used.

All other units to be discussed can be represented by the product of one of the units and a power of 10. For example, every possible unit of length can be developed by multi-plying the number of metres by the appropriate power of 10.

44

Table 2: Metric Units of Length

Name (Symbol) Metres

*kilometre (km) 103m or 1000 m hectometre (hm) 102m or 100 m decametre (dam) 101m or 10 m *metre (m) 100m or 1 m decimetre (dm) 101m or 1/10-m *centimetre (cm) 102m or 1/100 m *millimetre (mm) 103m or 1/1000 m

*preferred units

To make the system simpler the same prefixes are used with all units. For example, a millimetre (mm) is 1/1000 of a metre, a millilitre (ml) is 1/1000 of a litre, a milligram. (mg) is 1/1000 of a gram, etc.

According to the class, you may want to introduce the symbol 'm' for metre, 'cm' for centimetre, etc. The plurals, metres and centimetres, are also symbolized 'm' and 'cm', not 'ms' or 'cms.' Remember, these are symbols and not abbreviations and no period is used after a symbol.

Countries which have been completely metric for several years find that some terms, such as 'decimetre,' are not used in everyday living. People will talk of a book being 28 centimetres long rather than 2.8 decimetres long. You may wish to explain the term 'decimetre,' but it is not necessary.

Most people who feel that the metric system is complex are those who convert back and forth between the metric and English systems of measurement. When teaching the metric system, conversion to the English system is not necessary and should be avoided!

The metre is defined world-wide to be 1,650,763.73 wave lengths in a vacuum of the orange-red line of the spectrum of krypton 86. This is quite a definition! There are two reasons why such a complex definition was adopted:

1. the-length never varies and 2. this measurement can be replicated in laboratories

throughout the world.

From this brief history of the metric system it is hoped you

45

will take three main thoughts:

1. The metric system resulted from concentrated effort to develop a rational system of measurement. It did not develop haphazardly.

2. The problem of standardization has been solved in the metric system.

3. The metric system is both popular and useful because of its simplicity.

Activities

Experience and activity are key words in the teaching of measurement. Measure things! The success of this material will depend upon the amount of experience each participant has with the activities. The limited number of activities that are presented should stimulate possibilities for many more. Although the content is approached through activities and measuring experiences,, there is a need for exercises to further these experiences and to structure metric thinking. Two points should be emphasized:

1. It is important that you as well as your class do the activities in this section.

2. The activities will be more fun if done in a group situation.

Looking at Table 1 in the History and Rationale section, you will notice that you have to be concerned with only four base units. So, let's use the frontal attack, start right in on length, and begin inching our way down the metric road.

Length, Area, and Volume

In the groups where the metric system has been argued for years, there were two camps. One group wanted to use the centimetre, gram, and second for the core of the system and the other the metre, kilogram, and second. The latter group has prevailed.

It is strongly urged that first grade teachers not start with the metre. It is very difficult for first graders to handle a metre ruler. The same argument may be advanced for the kilogram and litre. Length will be approached as it should be covered with students, i.e., first measure with arbitrary units, then use the centimetre, next use the 10-centimetre (decimetre), and finally the metre. All measure-ment should be approached as a three step process:

46

1. Select a unit. 2. Partition the object to be measured into units. 3. Count the number of units used. That number is the

measure of the object.

Activity 1

Measuring objects with an arbitrary unit. Students should do several activities of this type using arbitrary units such as their thumb, a paper clip, pencil, crayon, cutout of their shoe, width of their. hand (a unit in the English system used for measuring the height of horses), cubit (another 'English' unit, the length of the forearm from the elbow to the tip of the middle finger), or other selected units. For your experience measure the chalk eraser, the width of your hand, the width of this book, and the length of a pencil using a paper clip as the unit.

In the illustration, a 'paper clip train' is being used to measure the width of a hand. Follow the three steps mentioned previously in the measurement process.

Record all answers. Then measure th again using pieces of paper the length of a thumbnail. Repeat the process measuring other objects. class emphasize four points:

1. The first unit should be lined up with the 'starting point' of the object.

2. The units should touch, but not overlap. 3. The 'train' should be straight. 4. The units should be 'rounded off' to the unit that has

its right end nearest to the 'finishing point' of the object.

In doing activities where arbitrary units are used, the need for standardized units becomes obvious. Ask several chil-dren to measure the same object, each with his own pencil. On the chalkboard, place their statements such as 'The table (or whatever object you pick) is S pencils wide.' 'The table is 7 pencils wide.' 'The table is 8 pencils wide.' Chil-dren will soon see that when pencils of differing lengths are used, different answers will result.

Activity 2

Developing the concept of a centimetre. Probably the first

In

47

metric unit the children will make use of is the centimetre. You will need (and each student in the class will need) 9 centimetre strips - 9 pieces of paper or cardboard 1 cm by 1 cm square.

The children, especially the younger ones, should have the experience of measuring many objects using centimetre strips. (If at the time you present this activity your students have studied two-digit numbers, have them measure objects longer than 9 cm.)

Using the centimetre strips, measure the length of a paper clip, a piece of chalk, the Cuisenaire 6-rod, the width of a hand, and the width of a thumb to the nearest centimetre. In this initial activity, actually use centimetre strips and not a ruler marked in centimetres. An exercise the children can do at their desks is to measure the pictures of objects drawn on a duplicator master. The pictures can be of pre-determined length. Measure the pictures below.

The arrow is about centimetres long.

The snail is about centimetres long.

The turtle is about .........centimetres long.

In exercises like these, the length can be controlled. Some answers should require 'rounding up,' and some 'rounding down.' The word 'about' is important in the sentence since

48

Li

a measurement is an approximation. As the children progress you can have them write not only the number, but also the name of the unit.

Activity 3

Measuring with centimetre rules. When the children have learned to use the centimetre strips in the measurement pro-cess, a ruler marked off in centimetres (not millimetres) should be introduced. It is strongly urged that the child construct his own 10-cm ruler during his first introduction to metric measure. He can do this by constructing a 10-cm train on a 10-cm long piece of paper, pasting the train on the paper, then numbering the cars from 1 to 10. Another approach is to construct a 10-cm ruler in front of the class. Then hand out 10-cm long pieces of paper already marked off in centimetres and have the children number the centimetres from 1 to 10.

The next few activities should involve the measuring of an object with a centimetre train, a 10-cm ruler, and finally with only a 10-cm ruler. When measuring an object with a 10-cm ruler work toward getting your students to 'read the ruler'-rather than counting the centimetres as they did with the trains.

In the example illustrated the child should learn to round off to the nearest centimetre and then read the ruler, 18 centimetres,' instead of counting 1 1, .2, 3, 4, 5, 6, 7, 8 centimetres.'

After the children have become skilled in using a 10-cm ruler, they should be given activities requiring them to

49

76

measure objects which are longer than 10 cm. When working with 5- and 6-year olds, be careful that the measure of the object is not a number the children haven't studied. In the activities concerning measurement it is the process that should be emphasized; the numbers themselves should never be a source of difficulty.

Now, using your 10-cm ruler, measure the length and width of this book and length of your forearm, the length of your foot, and length of your span (what is your span?).

Activity 4

The metre and notation. Initially, you may want to have your students measure objects with metre-long strips of un-marked cardboard. Then ask them to number the centimetres on the metre strip in groups of 10 using their 10-cm strips. Before proceeding further, have the class subdivide these cardboard metre rulers into centimetres. It is important that you do the activities with the same type of ruler your students will use. If you have a classroom set of wooden metre rulers, use one of them. Ideally, the rulers used should be marked off in centimetres, but if the ruler is marked off in centimetres (cm) and millimetres (mm) no harm is done. Measure the length, width, and height of your desk rounding off to the nearest metre.

The measurements for a desk, accurat to the nearest metre, might be 2 m long, 1 m wide, and 1 m high. Such measurements would not be helpful. The metre is used for much longer measurements, such as the length anc width of the classroom, the playgrou the school, the block, etc. To meas the dimensions of objects such as desks, tables, bookshelves, and peoç a metre ruler may be used and the results.recorded in centimetres. Pc example, a desk may be 152 cm long, cm wide, and 74 cm high.

50

You might say: I am 178 cm tall; what is your height (in centimetres)?

Just as 153 cents is written as $1.53, 153 centimetres is written as 1.53 metres. This can be interpreted as 1 metre and 53 centimetres which is read as 'one point five three' metres. Do not dwell on the mathematical use of the nota-tion - it is not necessary!

With your class, record the dimensions of your classroom, your desk, their desks, your height, and their heights in terms of centimetres, then in terms of metres using the decimal notation.

Activity 5

Area using arbitrary units. Here are some examples of area units:

Square

]

Thum Disc Blob Triangle \ kie Let the children give names to the units. Then follow the measurement process: select one of these units, match it against the area of some object, and count the number of units used. For example, the irregular figure below has an area of about 6 discs (if disc is the name given to the unit used). Emphasize that you are trying to 'cover' the object.

The units should be 'even with the edge' of the object, the units should touch, but not overlap, each other. Direct the children's attention to the parts of the object that are not 'covered.'

Make a cutout of some irregular area such as your thumb and make copies of it out of paper. Use your 'thumb' to find

51

the area of the top of a chalk eraser, of the irregular figure measured with the discs, of a cutout of your shoe, and of figure X.

Record the answers on the chalkboard in sentence form - 'The figure has an area of about thumbs.' Have your class perform similar activities.

Activity 6

Area using the centimetre square (cm 2 ). Have the children make centimetre squares (or have them available for use). The children should have the experience of finding the area of many objects.

Make duplicator masters for some areas that the class can measure with 'their centimetre squares. The figures below are 1 cm 2 , 9 cm2 1 25 cm2 , respectively. (reduced examples)

•ii U.'

You might point out that the square containing the 9 cm 2 has a side of 3 cm and the square containing the 25 cm 2 has a side of 5 cm.

Have the children use their centimetre squares to find the area of a stamp, a 10 cm ruler, the cutout of their thumb, the irregular figure which had an area of 6 discs, and figure X.

Activity 7

Volume, using the centimetre cube. In the initial develop-

52

ment of the concept of volume, it is important that children have the opportunity to construct several differently shaped objects each having the same number of volume units.

As with length and area, the study of volume should be introduced with activities making use of arbitrary units of volume, such as blocks, Cuisenaire rods, pencils, erasers, or even marbles.

Use 10 or 12 centimetre cubes in this activity. At first, let the children work on their own, constructing any objects they like. Encourage them to see that an object built of a specific number of cubes has a volume of the same number of cubes regardless of its shape. For example, the illustra-tion shows 4 different constructions, each having a volume of 4 centimetre cubes (4 cm3).

How many-differently shaped objects can be constructed with a volume of 8 centimetre cubes? When those possibilities have been exhausted, try the activity with 10 cubes.

Review: Length, Area, and Volume

1. Have your class compare the length of their feet, spans, and cubits. Why are these units useless as standard units?

2. Complete these statements. a. 128 cm =...m e. 1.06 m =_cm b. 108 cm =_m f. 10.01 m =cm C. 15 cm =.m g. 23.86 m =cm d. 101Ocm= m h. 0.09m= cm

3. What would be the length of the sides in a square containing:

53

a. 36 cm2-- cm? C. 4 crn2 - cm? b. 25 cm2-- cm? d. 16 cm2- cm?

4. How many different-shaped objects can you form with 6 centimetre cubes?

Capacity

Capacity can be thought of as the amount of material a con-tainer will hold. Capacity is usually linked to liquid measure though you may have already had your classes measure capacity by using sand to avoid using liquids.

In the metric system of measurement, volume and capacity are directly related. A container with a volume of 1 cubic centimetre (1 cm 3 ) will hold 1 millilitre of water. One milliltre (1 ml) is one thousandth of a litre (0.0019).

The need for fractional names such as 1/2, 1/4, 1/3, 2/3, 3/4, etc. will-diminish. The parts of the whole which need emphasis are - 0.1,. 0.2, 0.3 ... 0.9. Of course, in measurement, fractions could disappear completely, since 3/8 of a meter is 0.375 m or 375 mm. However, when working with the litre (the unit of capacity in the-metric system) don't worry now about using 3/49 , 2/3, etc. if it is the amount you want the children to see or work with. Since the metric system is based on 10 and since 1, 2, 5 and 10 are the only divisors of 10, we will probably talk about halves, fifths, and tenths of metric units. The decimal notation (112 is 0.5) will prevail eventually, even at the primary level.

Activity 8

Capacity and arbitrary units. The most obvious capacity units are handfuls. Give each child a container to fill with water or sand or other material you prefer to use. Have the children fill the container (milk carton, ice cream car-ton, cigar box, etc.) with 'handfuls' of material. Have them record their results on a piece of paper: 'My carton holds- handfuls of Compare the wide range of

54

results. Re-emphasize the need for a standard unit to measure capacity. If further experience is necessary, you may want to repeat the project with cups brought from home (since there are so many different sized and shaped cups). Try the activity yourself or get several containers such as an ice cream carton, a milk carton, a wastebasket, a big cooking pan, and a litre container. On a piece of paper write a pair of sentences for each container:

'The (name of container) holds about (guess) litres.' 'The (name of the container) actually holds (result) litres.'

In the first blank 'guestimate' the number of litres the container will hold. In the second, write in the results of measuring the object:

Don't forget the three step measuring process:

1. Select the unit - the litre. 2. Match the unit against the object - fill the object

using the litre. 3. Count the number of units (litres) used.

When the container is full (it is best to have a 'fill line' just below the top of the container) round off to the near-est whole litre according to whether more or less than half of the last litre was used.

Activity 9

Working with the litre. Get a container that holds a litre of water (and, ideally, has submarkings for each 100 ml). When you are collecting containers for your classroom, try to get as many different shapes as you can. It is important, especially in early experiences, that the children see that litre containers can come in many different shapes. It is the quantity the container will hold, not its shape that determines a capacity of 1 litre.

Once you get a litre container you can make many more. Pour a litre of water into a container and mark a 'fill line' for 1 litre on the outside with tape, or, if possible, cut the container so that it holds just 1. litre. Suggested existing containers which can be cut are quart, half-gallon, and gallon milk cartons, round quart, half-gallon, and gallon ice cream cartons. Containers that can be marked might be various shaped pans, cooking bowls, large tin cans, and bottles or jugs. Most activities for introducing the metric

55

units should be accompanied by some estimation exercises. Have the students estimate and record how many litres a container will hold, then measure the container to see about how many litres it does hold. Compare records.

Activity 10

Introducing the millilitre. The litre is a unit for capacity that is used for milk, gasoline, paint, and other quantities of considerable size. The litre is not used to measure small quantities, such as toothpaste, soda pop, medicines, frozen orange juice, etc. The unit used for the smaller measures is the millilitre (ml). If your school is going to get a set of metric capacity containers, try to get them in these sizes - U., 500 ml, 200 ml, 100 ml, 50 ml, 20 ml, and 10 ml. With such a set (whether bought, given, or contructed) one can do all the activities that are necessary.

Construct a container with a volume of 1 cubic centimetre (1 cm 3 ) to demonstrate the size of the millilitre (ml). Trace the figure below, then cut it out and tape it together along the edges. If you avoid spillage your cube will hold 1 ml of water.

''HI The children need several activities measuring the capacity of objects and recording the results in millilitres.. Have them first guess and then measure the capacity of a thimble, a match box, a tablespoon, and a teaspoon. Record the results in sentences like:

'I estimate that the thimble holds about 'It actually holds about ____ ml.'

Mass

As the metric system becomes the predominant system of measurement you may hear talk about the difference between mass and weight. A lunar example may be the best way to show the difference. Now that we are in the space age, practically everyone knows that a man weighs less on the

56

moon than he does on the earth. For example, a 300 kg man on earth would weigh about 50 kg on the moon, but he would have the same mass on the moon as he does on earth. Weight is dependent upon gravity, mass is not. Begin to stress the use of the correct metric term, mass.

The base unit of mass in the metric system is the kilogram (kg). For example, we say 'I have a mass of 78 kg.'

Activity 11

Arbitrary units of mass. To find the mass of an object you will need a balance and some arbitrary units such as paper clips, pencils, Cuisenaire rods, pennies, or other objects. Put a pencil on one side of the beam and then 'balance the pencil' with pennies (or multiples of any other small unit). Record the results on paper in a sentence like:

'The pencil has a mass of about pennies.' Repeat the activity with at least three other objects.

Activity 12

The unit used for small masses is the gram (g). This activity is very similar to the last. You will need gram masses. If you have a classroom set, that's great! If you don't, you can make one.

Put a gram mass on one side of the balance and balance it with a lump of clay or plasticine. Label your clay '1 g.' In a similar manner make a set of clay or plasticine 'masses' in multiples such as: 5 g, 10 g, 20 g, and 50 g. Use several small objects as test objects (a paper clip, a nickel, a penny, and a pencil). However, before you have the children put one of the test objects on the balance, ask them to estimate its mass in grams. Then find the mass of the object. Record both the guess and the result.

57

The quarter has a mass of about (guess) grams. It actually has a mass of (result) grams. Repeat the activity using other objects. Do you and the class get better at estimating mass?

Activity 13

Measuring mass using the kilogram. Hopefully, all schools will have metric scales available for finding the mass of children and other large objects using kilograms. For this activity, have each child find his own mass and then make and label a cutout of himself (perhaps using his projected shadow). Have him record his height and mass in metric units on the cutout.

Then you and your class might measure the mass of other objects, such as your own chairs, the textbooks used in the course of one day, litre of water (don't count the container - first find its mass when empty), a dictionary, and even the principal of the school (if he agrees). As mentioned earlier, there is a direct relationship between volume and capacity in the metric system of measurement. In fact, there is a direct relationship between volume, capacity, and mass. A container whose volume is 1 cubic cm (cm3) holds 1 ml of water and the 1 ml of water has a mass of 1 g. A container whose volume is 1000 cubic cm (or 1 cubic decimetre) holds 1000 ml of water (or 1 litre), and the water has a mass of 1000 g (or 1 kilogram). What did you get for the mass of one litre of water?

N.B. It is a good idea to label some of the objects in the room as you introduce each metric unit. For example, the aquarium may be 70 cm long, 40 cm wide, 35 cm high; have a water surface area of 2,800 cm', volume of 98,000 cm 3 ; a capacity of 98 of water and a mass of 12 kg. If the chil-dren label the objects as they study particular units, they will begin to think metric.

Review: Capacity and Mass

1. When finding the mass of something using a balance beam, how do you decide which unit to round off to?

2. Fill in the answers: a. 28 ml of water has b. 170 is ml. c. 3.12 kg is _____ g d. It would take

a mass of about _____ grams.

and 438 g or_____ kg. ml of water to balance 1 kg.

58

3. a. Will a car get a higher or a lower number of miles per litre than miles per gallon? (Is the litre larger or smaller than the gallon?)

b. Will a car get a higher or a lower number of kilo-metres per gallon than miles per gallon? (Is the kilometre longer or shorter than the mile?)

c. Gasoline consumption rates will be given in kilo-metres per litre. Will a car get a higher or a lower number of kilometres per litre than miles per gallon?

Tenrperature

This last section covers the introduction of a metric unit, the degree Celsius (°C), for which there is no physical model. On the Celsius scale for temperature, water boils at 100°C and freezes at 0°C. The unit is named after the Swedish scientist, Anders Celsius, who created the centi-grade temperature scale. The Celsius and centigrade scales are the same, but centigrade is no longer the proper term since the centigrade is a unit used to measure angles in the metric system.

The best way to get used to the Celsius temperature scale is to use it! It is almost a necessity that you have a Celsius thermometer. However, if you have a demonstration model of the Fahrenheit thermometer, you can rescale it using the nomograph shown here.

Activity 14

Graphing temperature's. Be sure to give the children lots of opportunities to read the temperature and record it in degrees Celsius (°C). Perhaps you could institute a morning weather report given by a different child each day to get the class to use Celsius thermometers and to give them a feeling for what the temperature is when expressed in degrees Celsius(°C). The previous day's high and low temperatures (taken from a newspaper account) could be recorded on a wall graph.

Review: Temperature

1. My body temperature is about ____°C. 2. Normal room temperature is about °C.

3. Water boils at about °C. 4. A warm summer day would be about ____ °C. S. The temperature in a refrigerator is about °C.

59

6. The temperature in the vegetable section of a super-market is about °C.

7. Water freezes at about ____°C. 8. What temperature on the Celsius thermometer is the same

on the Fahrenheit thermometer? °C.

YJ

100 -- --- water boils

90 ----- /99

80 ----- I76

70 - - - -- 15g

(DO -- ---

bode emperaur 30 -----

2O 62

10 _____ 50

32 water

-30

oc Fahrenheit Materia is

At the beginning of metric conversion, many schools will have a problem gathering supplies. Certain materials are necessities in the teaching of measurement and metric measurement is no exception. Fortunately, most of the materials are inexpensive or easily constructed. In the section on length, the construction of some of the rulers is discussed. If you have one metric ruler, you can con-struct the rest. If you have one metric ruler, you can also contruct the centimetre squares and cubes needed for the study of area and volume.

The construction of units of capacity and mass have also been discussed. When it comes to temperature you should

60

have a thermometer available for classroom use. If it is a Fahrenheit thermometer, you should rescale it to degree Celsius using the nomograph given earlier.

Following is a list of companies and government agencies that are currently producing materials or can give some assistance with this problem of teaching the metric system of measurement.

Addison-Wesley (Canada) Ltd. - Don Mills, Ontario Buntin Gillies Co. Ltd. - Ottawa, Ontario Cameron Products - Bramalea, Ontario Canadian Metric Association - (P.O. Box 35) - Fonthill,

Ontario Contrasts 20 - Calgary, Edmonton, Vancouver, Winnipeg,

Regina (Nearest Barber-Ellis Office) Kruger Pulp and Paper Ltd. - Moncton, Toronto, Hull, Montreal (Nearest Office)

Information Canada (Under Government of Canada) (Nearest Office)

Jack Hood School Supplies Co. Ltd. - Stratford, Ontario Lufkin Rule Co. of Canada Ltd. - Don Mills, Ontario Lily Cups Ltd. - Scarborough, Ontario MacLean-Hunter Learning Materials Co. - Toronto 101, Ontario Metric-Aids Ltd. - Toronto, Ontario Moyer-Vico Ltd. - Moncton, Weston, Winnipeg, Saskatoon,

Edmonton, Vancouver and the Longueuil Co. in Chambly (Nearest Office)

The National Council of Teachers of Mathematics - 1906 Association Drive, Reston, Virginia 22091

Sargent-Welch Scientific Co. of Canada Ltd. - Weston, Ontario

Spectrum Education Ltd. - Toronto, Ontario Spicars International Ltd. - Scarborough, Ontario Toronto Dominion Bank (Nearest Office)

61

NUMBER THEORY (A Grade 8 Introductory Lesson)

Jim Ludgate, Campbell River Junior Secondary School

The following lesson was developed in an attempt to intro-duce evens, odds, primes and factors in a way that would be fresh, perhaps stimulating. I approached the use of 'counters' with some reservations, but was amazed at their positive reception.

Equipment

approximately 20 squares or cubes. display device such as overhead projector.

Procedure

1. Define a 'rectangle': an array of three or more squares containing at least two rows and two columns such that the perimeter of the array forms exactly four right angles. This may be presented by example, or verbal definition. Precision is not essential, as students catch on to the basic principle quickly.

2. Begin a table: Fill in the first column up to 8 at first. Now begin forming arrays on the screen asking for suggestions as to how a rectangle might be arranged. You could introduce the idea of two by three and three by two arrays now, or wait until later.

No. of squares rectangle?

3

4

S

6

7

8

3. When 3 to 8 have been done, stop and ask for a rule or principle that might be used to predict which numbers will produce rectangles. Try it yourself before looking at step four.

4. Now extend the table to 13 or beyond. Using 9 squares, solicit arrangements from the class. Some leading may

62

be necessary to break them from even numbered rows. Continue to past 13 and ask for a general rule.

Beyond this point, I found that the lesson directions. Some classes will not be able numbers. I have taken these students back column of the table and starting at 4, dem arrangements possible for each rectangular them to direct my arrangements.

can go in various to identify prime to the third Dnstrated all the shape, allowing

No. of squares

Rectangle? Arrangements

3 X

4 V 2x2

5 X

6 2x 3, 3 x 2

7 X

8 1 9 1

10 1 11

12

If you do this, it may even be useful to suggest incorrect arrangements, forcing students to think through this physical model of the concept of factor pairs. This leads easily into informal definitions of primes.

When a class picks up the prime number patterns quickly, the above procedure may be a valuable addition to the lesson in laying a conrete basis for the 'factor' concept.

And when you assign seat work involving factors, be prepared for requests to use your squares. Some students may have missed the chance to perform concrete operations earlier in their development.

63

NEW METRIC MATERIAL

Bill Kokoskin

1. Metric Material available from:

Bowater Canadian Ltd. P.O. Box 1000 Station 'A' Toronto, Ontario M5W 1G5

2. The Toronto-Dominion Bank has produced a set of four posters related to metrication. One set of these posters will be sent to each elementary and secondary school in the province. Additional sets are available, free-of-charge, from the Bank. Requests for sets of posters should be directed to:

The Toronto-Dominion Bank Pacific Division, Toronto-Dominion Tower Pacific Centre, P.O. Box 10001 VANCOUVER, British Columbia V7Y lAl

3. Material available from the Department of Education, Victoria, B.C.

Introduction to the Metric System (Federal Government Metric Commission)

This booklet will be available on the basis of one copy for each elementary and secondary school teacher. It will be provided to schools upon request from the Curriculum Resources Branch. The normal supplementary requisition form should be used. Orders may be placed now and the booklets will be distributed as soon as stock is received.

4. A METRIC HANDBOOK FOR TEACHERS

.edited by Jon L. Higgins, is a joint project of the NCTM and the Educational Resources Information Center (ERIC) Clearinghouse for Science, Mathematics, and Environmental Education. Contributions by seventeen authors have been compiled in this 144-page handbook to provide practical suggestions for teaching the metric

64

system.

The articles - some reprinted from recent issues of the ARITHMETIC TEACHER, some written especially for this publication - are divided into five sections: 'Intro-ducing the Metric System'; 'Teaching the Metric System: Activities'; 'Teaching the Metric System: Guidelines'; 'Looking at the Measurement Process'; and 'Metrication, Measure, and Mathematics.'

The book sells for $2.40, with discounts on quantity orders shipped to one address as follows: 2-9 copies, 10%; 10 or more copies, 20%. Make cheques payable to the National Council of Teachers of Mathematics.

NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS 1906 Association Drive, Reston, Virginia. 22091

5. A. CS/I Standard Z234.2 - 1973 'The International System of Units (SI)'

Approved by the Standards Council of Canada as National Standard of Canada No. CAN-3-001-01-73

This Standard, based on the document 'Le Systeme Inter-national d'Units (SI)' published by the International Bureau of Weights and Measures, contains definitions of SI units and Resolutions and Recommendations of the General Conference of Weights and Measures.

Available in English as well as in French for $2.50 per copy.

B. CS/I Standard Z234.1-1973 'Metric Practice Guide'

Approved by the Standards Council of Canada as National Standard of Canada No. CAN-3-001-02-73

This Standard is the primary source of information con -cerning the application of SI. It contains a section based on International Standard ISO 1000-1973 'SI units and recommendations for the use of their multiples and of certain other units' and sections dealing with con-version factors which relate to the SI and yard/pound systems, guidance on rounding of data and dual dimen-sioning.

Now available in English for $4.00 per copy. A French edition will be available shortly.

65

Order from: Canadian Standards Association (Standards Sales), 178 Rexdale Blvd., Rexdale, Ontario. M91V 1R3

Orders of $5.00 or less must be accompanied by remittance.

NEW BOOKS ACROSS MY DESK

Bill Kokoskjn

1. Sow to Think Metric by W.W. Bates and Olive Fullerton, Copp Clark Publishing, 1974, 73 pages.

Available from Moyer Vico Ltd., Canadian Library Supply Company, #205 - 1899 Willingdon Avenue, Burnaby, B.C., V5C 5R3 at $1.50.

2. World of Metric Activity Workbook by Sharon F. Odegord, 102 pages.

Available from Moyer Vico Ltd., Canadian Library Supply Company, #205 - 1899 Willingdon Avenue, Burnaby, B.C. V5C 5R3

3. Pre-Calculus Mathematics by Shanks, Brumfiel, et al, Second Edition. Addison-Wesley, Don Mills, Ontario, 1972. 628 pages.

4. Success with Mathematics by Eicholz, et al, Addison-Wesley, Don Mills, Ontario, 1974, 408 pages.

5. Introductory Algebra - A Modern Approach by Keedy Bittinger. Addison-Wesley, Don Mills, Ontario, 1972, 485 pages.

6. Mathematics for Living by Norman Frome. Macmillan of Canada, Toronto, 1972, 323 pages.

7. Algebra I by Smart, Rogaisky, Ruehmann. Ginn E Co., 1974, 533 pages.

8. Algebra 2 and Trigonometry by Doluioni, et al, Hougton-Mifflin, 1974, 628 pages.

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NOMINATIONS TO BCAMT EXECUTIVE

Would you please send your nominations by JANUARY 31, 1975 to

ROGER SANDFORD.

The offices vacant are:

1. PRESIDENT

2. SECRETARY

3. PUBLICATIONS CHAIRMAN (Editor of Vector)

All nominations should have the permission of the person nominated who should be a member of the BCAMT as should be the nominator.

SEND NOMINATIONS TO:

ROGER SANDFORD R.R. #1 TZOUHALEM ROAD DUNCAN, B.C. V9L 1M3

PSA74-132/cw 67