BRITISH COLUMBIA ASSOCIATION OF MATHEMATICS TEACHERS

NEWSLETTER/JOURNAL

VOLUME 15, NUMBER 1 NOVEMBER 1973

BCAMT EXECUTIVE 1973-1974

Past President J. Michael Baker 11225 - 87th Avenue Delta 594-8127 (home) 588-1258 (school)

Vice-president Roger Sandford R.R. #1, Tzouhalem Road Duncan 746-6418 (home) 749-6634 (school)

Treasurer Bill Dale 1674 Tull Avenue Cou rtenay 338-5159 (home) 334-2428 (school)

N.W. NCTM Conference Organizer R. McTaggart 4780 McKee Burnaby 1 434-0716 (home) 736-0344 (school)

NCTM Representative Tom Howitz 2285 Harrison Drive Vancouver 16 325-0652 (home) 228-5203 (UBC)

In-service Specialist Dennis Hamaguchi MacDonald Park Vernon 542-8698 (home) 542-3361 (school)

President Alan Taylor

7063 Jubilee Street Burnaby 1

434-6315 (home) 936-7205 (school)

Secretary Miss Florine Katai 866 Cantley Road

Richmond 277-8524 (home)

943-1105 or 943-1106 (school)

Publications Chairman Bill Kokoskin

1341 Appin Road North Vancouver 988-2653 (home)

985-3181 (school)

Elementary Representative Mrs. Grace Dilley

2210 Dauphin Place Burnaby 2

299-9680 (home) 588-5918 or 581-0611 (school)

Secondary Representative Mrs. Carryl Koe

5775 Yew Street Vancouver 13

266-9916 (home)

Primary Representative Mrs. Alice Hayman

11 - 8747 Granville Street Vancouver 14

261-2990 (home)

The B.C. Association of Mathematics Teachers publishes Vector (combined newsletter! journal). Membership in the association is $4 a year. Any person interested in Mathema-tics education in B.C. is eligible for membership in the BCAMT. Please direct enquiries about membership or articles to be published in Vector to the Publications Chairman.

Inside This Issue

1 A Message from the President A/an Taylor

2 The Mathematics Summer Workshop - A View from the Bridge Dr. Walter Szetela

5 Correlated Curriculum Guides Alan Taylor

6 Compounding the Interest of a Mathematics. Workshop Dr. Walter Szetela

11 Right and Wrong Answers Dominic A. Alvaro

13 Lesson Plans Oscar Schaaf

17 BCTF Lesson Aids

21 Football Poker S.G. Bell

25 Book Reviews Roger Sandford P. Peak

27 The Daignost,ic Prescription Clinic UBC Mathematics Education Department

28 New Books Across My Desk Bill Kokoskin

SHARE Your ideas are wanted and needed by your

colleagues. You must have one favorite

lesson that you know is different and goes

over well with a class.

send it in The lessons don't have to be written in detail

either - perhaps something like the FOUR

that appear in this issue by Oscar Schaaf.

A Message from the President In the first issue of Vector for this year I outlined the major objectives of our association. The primary aims revolved around communication and in-service. Further to my comments at that time, I wish now to elaborate on the action taken to date.

1. Communication - a two-way street Horizontal lines of communication between teachers and the BCAMT are links vital to the effectiveness of our specialist association. We attempt to make contact with you through the publication of Vector and the organi-zation of workshops. In turn, feedback from you is necessary to estab-lish our direction.

You are welcome to attend executive meetings, to get in touch with us by telephone or letter and to submit articles to Vector. To facilitate involve-ment outside the Lower Mainland, we are participating in meetings in other areas of the province. The first such meeting was in Campbell River on November 2. We were invited to take part in a workshop spon-sored jointly by the Courtenay and Campbell River mathematics teachers.

To establish more direct lines of communication between the BCAMT and the Mathematics Revision Committee, we are pressing for the appointment of two observers to attend the committee's meetings. To date, we haven't been successful. We do, however, intend to pursue this objective.

2. In-service We encourage the establishment of chapters. To date, there are several active groups in the province. To encourage this necessary activity, we shall be publishing suggestions on how to establish a workable organiza-tion at the local level.

Dennis Hamaguchi, from Vernon, has been appointed to the executive as In-service Representative. He would appreciate hearing from you on mat-ters of this nature.

A summer workshop is being discussed at this time. Plans are still in the preliminary stage, but we expect such a workshop to center on revision at Grades 4, 511 6 1 9 and 10 levels.

3. 12th Annual Northwest Mathematics Conference The conference held at the Hotel Vancouver on October 5-6 was a tremen-dous success. Congratulations on an excellent job are extended to the

conference committee. A report on the conference is expected in the next issue of Vector.

4. New Appointments to the Executive We are pleased to welcome two new members to the executive. Alice Hayman has been appointed to the position of Primary Representative, and Carryl Koe has received an appointment as Secondary Curriculum Representative. These appointments were designed to provide a more complete service to our members.

- Alan Taylor

The Mathematics Summer Workshop

- A View from the Bridge by Dr. Walter Szetela

Last year, Jim Sherrill was wagonmaster through uncharted territories as he guided his train to its destination, the first Summer Conference held by the BCAMT. Spurred on by the success of that endeavor, the BCAMT plunged boldly forward, moving confidently over territory more familiar now, with plans for a Mathematics Summer Workshop.

Although the committee was formed late, it was extremely active, and there was no dearth of ideas, proposals and suggestions for the workshop. Marian MacDonald sent the committee off and running with a long list of potential presenters of primary workshops. Unfortunately, it seemed that many of those people were going to Europe to spend their dollars before they were further devalued. However, through the persistence of the com-mittee, a very able nucleus of teachers, supervisors and professors was re-cruited, giving high hopes that eventually a wide variety of workshops could be offered at all grade levels. As the number of highly qualified and enthusiastic workshop presenters grew rapidly, the committee experienced some periods of anxiety about the prospect of having more workshops available than people in attendance. Perhaps most people have a tendency toward procrastination, but teachers are notorious. In the first few weeks fol-lowing notice of the workshop, several cobwebs were seen near Berniece Fender's desk in a corner set aside in the BCTF office for processing pre-registrations. One thing is certain; no cobwebs were growing on Berniece, who was providing a tremendous service in preparing brochures for this workshop and many other BCTF-sponsored summer courses and generally

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satisfying innumerable requests from the committee. Not until the last two weeks preceding the workshop was it apparent that there would be an excellent attendance, for registrations began to pour in. From that time on, the response and participation was gratifying.

Following a hectic period of on-site registration the first morning of the workshop, in which registrations reached about 300, the workshop began in earnest with Eric MacPherson's presentation on 'New Directions in Mathematics.' The Auditorium was barely able to accommodate the audience (with many standees) listening to Eric present various avenues to the teaching of mathematics, with their caveats and limitations as well as their advantages.

After the keynote address, teachers immediately headed for workshops in their particular areas of interest. Enthusiasm ran high, as registrants flocked for choice seats, which, in many cases, were not long available. Every session was at capacity or greater, and Lorna Robb, who was every-where taking pictures, taping presentations, absorbing everything during the workshop, counted 125 persons crowded into one classroom at Kathy Mclnally's first presentation. Kathy, a committee member herself, reacted like a trouper in the sardine-packed surroundings and presented her stimu-lating and practical workshop on primary math games.

At each session, there were usually seven workshops, with at least one workshop for primary, intermediate, junior secondary and senior secondary levels. It was apparent that the greatest interest and attendance came from the primary teachers, who came early to each session, participated zealously in each workshop activity, and came out beaming, obviously happy in their knowledge of new and practical ideas and learning activities.

Betty Huff gave two workshops on geometry activites. With attendance limited to 36 1 she had to put up a 'closed' sign well before the workshops were to begin. One of the few criticisms heard about the workshop was the lack of larger rooms, which not only made participation by all atten-dants impossible, but also hampered the presentations of those giving the workshops.

Iry Burbank's workshop on use of audio-visual materials and multi-media for intermediate grades was extremely well-attended and well-received; it provided an excellent lead-in for the continuing demonstration of equip-ment and materials by Don Lyons of Burnaby Resource Center. Adjacent to the equipment demonstration room was the Ideas Room that Carryl Koe and Hugh Elwood had set up. Teachers could and did browse through the many exhibits arranged by Carryl and Hugh. They were

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observed forming exotic soap bubbles and writing down minute details of games that struck them as particularly useful for their students. The room was a kaleidoscope of color and imaginative exhibits, in a sense a work-shop itself, a static display that became dynamic as soon as a registrant manipulated a device or played a game.

The variety of workshops is exemplified by such presentations as Dave Gemmell's Geodesic Domes, John Klassen's and Mike Baker's valuable sug-gestions for using multiple texts, Leo Rousseau's demonstration lessons at both primary and senior secondary levels, and Brenda Leeson's and Roger Freschi's Open Area Math Stations. Dave Robitaille and Sid Linstedt both presented sessions on problem-solving; Jim Sherrill, Ken Pelling and Larry Evans made many teachers more comfortable with the metric system. Detailed examinations and evaluations of new textbooks in connection with mathematics curriculum and objectives were given by Heather Keliher, Rob McTavish and Roy Lister.

Bruce Ewen seemed to be everywhere on the program with presentations on calculus, logarithms, and as part of a secondary school teachers' panel on the changing role of the secondary school teacher, As expected, Dominic Alvaro, Dan Shimizu, Bruce Jordan, Anita LoSasso and Ewen found some areas of disagreement, which helped to spark the discussion with some needle-like questions from the audience.

David Fielker not 'only had something to say about teaching mathematics, but also his clever English wit was enjoyed by the audience. Tom Howitz demonstrated the uses of labs and activities. lisa Link and Karen Susheski gave excellent presentations on operations in primary grades. Doug Owens and Werner Liedtke presented different aspects of measurement in primary grades. Many other workshops and people not mentioned provided teach-ers with hundreds of ideas and demonstrations of inestimable value.

There were a few anxious moments during the workshop when some speakers failed to show up well ahead of time. On a casual inspection of various rooms, I noted a room swarming with teachers at 1:25, with no speaker in sight for a 1:30 workshop. As the sweat formed a puddle on the floor, in waltzed Lynn Leluck, beaming, at 1:28. It was an interesting way to meet Lynn, whose warmth and personality on the telephone gives only a hint of her captivating charm in person. But Lynn, could you arrive five minutes early next time?

There were some headaches, a lot of telephone time and letters, but mostly there was tremendous satisfaction in seeing such a large and enthu-

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siastic reception for the workshop. It is unfortunate that the Curriculum Revision Committee was unable to attend the workshop to see, feel and hear what mathematics teachers at all levels are doing.

After two very successful summer meetings, future chairpersons should feel even more confident that B.C. teachers are eager to support well-planned and appropriate programs. However, any such endeavor must, rely heavily on a committee that is willing to spend much time and effort on the multitude of problems such an undertaking involves. It was most for-tunate that this year's committee contained persons with many different strengths as well as the desire to work. Although every member of the committee contributed, special credit must be given for the valuable con-tributions of Marian MacDonald, Carryl Koe, Hugh Elwood and Alan Taylor. Each one of them was a vital organ in this body. Finally, one cannot overlook the contributions of those people who always said yes when needed, Grace Dilley and Florine Katai, for example.

Correlated Curriculum Guides by Alan Taylor

Numerous requests for information about the new mathematics textbooks have been received. As you are aware, the Department of Education pro-vides a curriculum guide for each course. The adoption of a 'multiple-textbook' approach, however, creates a demand for guides that correlate topics in the textbooks at each grade level.

Several groups in the province have been active in producing correlation guides for their districts. I have attempted (with their permission) to pro-vide BCAMT members with a list of some of the guides produced. If you are aware of additional guides, please send the information for publication.

The following correlated guides are available to interested teachers:

Grade(s) Level Guide Author(s) To Order

K-3 Primary Mathematics topic - correlations

Marnie Buchan Vicki Kidol Carol Gibson Janet Sammon Linda Murtsell Colleen Walton Terry Straumford Janet Wilson Heather Kelleher Marg Oswald Mary Schumacher

Contact Mrs. M. Oswald Primary Consultant, New Westminster School Board, 821-8th Street, New Westminster (a limited number are available free of charge)

4 Grade 4 - A Correlated Ken Pelling\ Contact: Stan Heal, Principal Curriculum Guide Larry Evans \ Courtenay Junior Secondary - based on the course to be J 700 Harmston Avenue introduced in September, 1974. J Courtenay, B.C. - a correlated curriculum guide including a general analysis of / Submit 504 for each copy ordered each text. to cover substantial paper and

\ mailing costs (Make cheques payable to

7 Grade 7—A Correlated Ken Pelling \ School District No. 71) Curriculum Guide Larry Evans (see above comments)

7 Mathematics 7 - Course Ralph Gardner Objectives and Correlation of Topics with Page Numbers

8 Cross Reference for School Stan Heal Math 11, Mathematics 11 and Essentials of Mathematics 2

Contact: Ralph Gardner, Super-visor of Intermediate Years, School District No. 43 550 Poirier Street, Coquitlam, B.C. (a limited number available free of charge)

Contact: Stan Heal (see above instructions)

Com poundin g the Interest of a Mathemati cs Workshop by Dr. Walter Szetela

The 300 teachers who participated in the Mathematics Summer Workshop at Centennial School, Coquitlam this summer received much in the way of ideas and practical suggestions for the mathematics classroom, as well as renewed confidence and enthusiasm. These teachers, however, represent only a small fraction of the teachers in the province who are confronted with instructional problems of increasing complexity as they attempt to fit new textbooks, materials, curriculum and methods together. These teachers are anxious and willing to improve their mathematics instruction and are not averse to using good ideas when such ideas are made known to them.

Where can they find these good ideas and suggestions for better mathe-matics teaching? Obviously, they could have found some of them at the Mathematics Summer Workshop. Unfortunately, many teachers were un-able to attend that workshop. The people who presented workshops col-lectively represent a reservoir of knowledge and experience that should continue to be used rather than lie dormant. Putting the know-how of these people to rest is like stuffing money in a mattress. Money and

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ideas diminish in value through disuse, but may result in large dividends when properly used. The BCAMT urges school districts throughout the province to take advantage of its human resources.

To encourage and expedite such use, a preliminary list of workshops, with the names and addresses of persons who have presented such workshops, is given below. The people whose-names are listed responded positively when invited to become part of a pool of workshops from which school districts could select appropriate programs to answer particular needs of their teachers. These workshops deal with specific problems and are limited in scope. They certainly do not provide an answer to the vast multitude of problems, some general, some unique, within the various school districts. But they may provide a beginning or encourage increased dissemination of ideas, which may help to overcome some problems in mathematics education.

In addition to the particular workshops listed below, a considerable num-ber of people have been and are currently available to present workshops on various topics at different grade levels. Sandy Dawson, in-service chair-person for the past two years, has compiled a list of these people that has been circulated throughout the province and is available at the BCTF office.

Dennis Hamaguchi has been appointed In-service Specialist for the BCAMT and will be endeavoring to expand the opportunities for in-service work-shops as well as maintain an up-to-date list of available personnel. If there is a particular need in a school district that is not covered by an available workshop, districts are invited to describe the need and to write to Mr. Hamaguchi, who will either locate a suitable workshop presentation or ask appropriate persons to design a workshop to satisfy the need. This publication can also serve as one means of communicating descriptions of workshops. Such descriptions should be sent to Mr. Hamaguchi. (See inside cover for his address.)

The willingness of people to present workshops is commendable. However, all of them have a primary responsibility within their own districts or in-stitutions and are not available at all times. District administrators inter-ested in the services of these people should communicate with them to arrange workshops at mutually agreeable times. Districts would probably be expected to reimburse travel expenses and provide an honorarium.

1. Primary Math Activity Program This workshop makes use of teacher-made and commercial materials, original work cards, and simple, effective games. The organization of such

a program is explained, and ample 'make and take' time is provided. It is suitable for Grades 1-4.

Ms. Valerie Bortoletto 4228 Victoria Drive Vancouver 12 1 B.C.

2. Games and Activities for Primary and Intermediate Grades This workshop is intended to add more fun to the learning of mathema-tics. It would include games, activity corners, bulletin boards, construc-tion and application.

Roy Brock 2475 West 16th Avenue Vancouver 9 1 B.C.

3. Using and Making Audio-Visual and Multi-media Materials in Teaching School Mathematics The workshop consists of two parts. The first part consists of a brief illustration of how the overhead projector, audio tapes, slide projector, camera, synchronized sound tape-slide systems, film strip projectors, etc., can be used effectively in teaching mathematics. Part two consists of teachers making, developing and having hands-on experience in the use of these materials.

Dr. Irvin K. Burbank Faculty of Education University of Victoria Victoria, B.C.

4. An Experiential Approach to Calculus An introduction to calculus that depends upon a student's investigation of the 'slope' of nonlinear functions. He/she uses ideas with which he/she is familiar, rather than a set of definitions.

Bruce Ewen 1606 Eastern Drive Port Coquitlam, B.C.

5. A Practical Approach to Logarithms Fundamental ideas are put together to permit a student to make his/her own log tables, then use them for multiplication, division, powers and roots. It is devised for 'nonacademic' use, but is useful for all ability levels.

Bruce Ewen 1606 Eastern Drive Port Coquitlam, B.C.

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6. The Place of Labs and Activities in Achieving the Objectives of Math 7 and 8 The basic objective of this workshop is to help teachers construct their own labs and activities. The characteristics and uses of labs will be exam-ined. Suggested activities from the newly adopted texts will be expanded to attain specific objectives. Labs and activities will be stressed as one of the many strategies a teacher can use successfully. The workshop will illustrate how little equipment is needed to start a mathematics laboratory.

Dr. Thomas Howitz Faculty of Education University of British Columbia Vancouver 8, B.C.

7. Preparation and Application of Learning Packages in Mathematics Technical aspects of preparing learning activity packages and their intro-duction into the classroom with emphasis on the revised Math 8 program. A set of packages of this course is available for examination and discussion.

John Klassen 1830 Greenock Place North Vancouver, B.C.

8. Measurement in the Primary Grades Suggestions will be given for an instructional sequence (problems, activities, games) for various measurement topics (i.e., length, area, capacity, time).

Dr. Werner Liedtke Faculty of Education University of Victoria Victoria, B.C.

9. Primary School Math Games Practical tips on making and using math games in the primary grades are given. Numerous games are displayed and used to illustrate the sugges-tions.

Ms. E.K. Mclnally Associate in Education do Simon Fraser University Burnaby 2, B.C.

10. A Practical Look at the Metric System The workshop introduces the metric units that will be in common use and involves a series of stations for participants to obtain experience in work -ing with metric units. The aim of the workshop is to reduce fears and apprehensions about introducing the metric system in schools. Larry Evans, R.R. #1 or Ken Pelling, Comox Airport Comox, B.C. Courtenay, B.C.

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11. A Correlational Evaluation of the New Grade 7 Math Program The correlation of topics from the new Grade 7 math texts with the Grade 7 curriculum as evaluated by Larry Evans and Ken Pelling in a pro-ject over a five-month period. They discuss the strengths and weaknesses of the new texts in the context of the B.C. curriculum. Larry Evans or Ken Pelling R.R.#1 Comox Airport Comox, B.C. Courtenay, B.C.

12. Introduction of the Metric System in Primary Grades This workshop consists of four parts, as follows: 1. Discussion of efforts and problems encountered thus far in converting Canada to metric system. 2. Presentation of metric units and suggestions for teaching. 3. Question period. 4. Experience and practice with metric units at stations throughout room.

Dr. James Sherrill Faculty of Education University of British Columbia Vancouver 8 1 B.C.

13. Teaching the Whole Numbers With Sums From 10 to 18. One-third of the workshop describes approaches used by the new text-books. The other two-thirds consists of games and activities that can be used to teach these particular sums.

Mrs. Karen Susheski 11145-86 Avenue Delta, B.C.

PROFESSIONAL DEVELOPMENT

Any BCTF member who wishes to receive the Pro. D.

Informational Bulletin this year, should send his/her name,

address, social insurance number, school name and School

District No. to:

B.C. Teachers' Federation 105 - 2235 Burrard Street Vancouver, B.C. V6J 3H9

This is a service to members of the BCTF.

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Right and Wrong Answers by Dominic A. Alvaro

During a student's search for an answer he/she may, in certain instances, seek approval or disapproval. Whether the approval comes from the teacher or his/her peers, expression such as 'I agree' or 'I disagree' should be used in preference to 'right' or 'wrong.' When a student gives a re-sponse to a question, his/her response could be the result of a 'game' different from what the teacher or other members of the class are playing. It could be that the student's response is just as appropriate as that of the teacher, but perhaps the student's assumption of different rules is giving him/her a different response to the same question. If this be the case, it becomes a matter of the student's noting his/her use of different rules, and changing his/her rules if he/she wishes to conform and play the game the teacher and/or the rest of the class are playing.

To illustrate, let us examine the following problem,

1 +5_ 3 9

Suppose the student responds with - , adding the 1 to the 9 and the 3 to the 5 by confusing division. The teacher might then ask 'Johnny' - any student - to explain how he arrived at the 'answer.' Johnny might resort to counters, pebbles or colored rods to explain his answer, but in so doing is most likely, of course, to fail. He will find an inconsistency with the physical world. On the other hand an alternative might consist in posing to

Johnny the further question:

What is-i- + -- ?

If Johnny is consistent in playing his 'game' and responds with the teacher could proceed in this fashion:

Teacher: 'add 3 to 5' Johnny: '8' Teacher: 'add 5 to 3' Johnny: '8' Teacher: Do you wish fractions also to yield the same sum when added

in reverse?

If Johnny answers, 'yes,' the teacher will confront him with the lack of

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commutativity in his system. Should the answer be 'no,' there is still at least one other argument that can be presented to him.

Teacher: What is2- +-L = ? Student: L2 (if he is still playing the same 3 10 4 game)

At this point the teacher can ask for the standard name for14 which might be given as 3. Johnny now is faced with accepting a game that yields such a result as: -) £.. +_L = 3

3 10 It is very likely that Johnny himself will accept this conclusion for his own reasoning. He therefore rejects his present game as workable for adding fractions!

Returning to the previous illustration, in which Johnny was persuaded to apply his own game in a physical context, it does not follow that this approach will always be inapplicable.

Consider another possible response: 1 + = 3 9 12

Johnny and the teacher know that he has added numerators and denomina-tors - a common 'mistake' by students at the junior secondary level. Pushed on to a physical interpretation Johnny might say:

represents the games won per games played by the Montreal Canadians hockey team at home during the first week of schedule.

the team's away record for the next three successive weeks. How would one express the team's record at the end of the first month?' Clearly,

would do just that.

Indeed, it is highly improbable that a student, say, in a Grade 8 class working on fractions would give-9- as the 'answer' to the problem - + -. = , and then claim that he/she is calculating the won/per game record of the Montreal Canadians hockey team, but it is salutary for the teacher to know that it is a possibility.

In summary, all examples above point out that it may be a better learning situation for the teacher to suspend judgment of 'right' or 'wrong" until he/she finds the reason behind the student's written work. Furthermore, it becomes more meaningful for the student if the teacher helps the student see for himself/herself whether he/she is 'right' or 'wrong,' although the teacher must also be aware that it is not always possible for him/her to facilitate discovery for the student.

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Lesson Plans Four lesson plans are provided in this issue. Oscar Schaaf distributed these at the 12th Annual Northwest Mathematics Conference. Do you have any that are as good?

LESSON PLAN NO. 1

AN INTERESTING GEOMETRY ACTIVITY Grades 4 to Adult

1. Draw a large quadrilateral with all sides of different lengths. 2. Find the middle point of each side. 3. Connect the mid points as shown in the figure.

4. Cut out the large quadrilateral. 5. Snip off the regions A, B, C and D. 6. See if you can piece A, B, C, and D together (like a jigsaw puzzle)

completely inside region E. 7. Will this always work? Test by repeating the experiment. 8. Repeat steps 1-4 with a large square. 9. Can you fold regions A, B, C and D over so as to completely fill

region E? 10. Would this folding method work for any rectangle? any diamond?

any paralelogram? any kite? any trapezoid? certain trapezoids? 11. What must be true of a quadrilateral if the folding method is to

work?

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LESSON PLAN NO. 2

1. Make a star: a. that has edges each 2" long and b. points that are angles of 2011 and where c. the openings between the arms are angles of 800. d. How many points does the star have?

2. Make a star: a. where the edge is 2", the inner point angle (point) angle is 20 0 and

the outer angle is 1101. b. How many points does it have? C. Record this data on the table below

3. a.b. C.

d. e.

Make stars with edge 2" where the point 4 = 200 outer = 920 point 4 = 300 outer = 1020 point . = 300 outer - = 900 point . = 200 outer z. = 710 point = 300 outer - = 81°

Construct this design

4. Can you make a star with 2" sides and using a compass and

a. 8 points and where the point = 2002straightedge.

b. 8 points and where the point - = 3007 What is the outer angle?

5. Record the indicated information in the table for each star you made. Search for a relationship.

Point Outer Z Number of Points

b. Can you make other stars of your own?

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B A

LESSON PLAN NO. 3

CONSTRUCTIONS AND RELATIONSHIPS Given Trapezoid ABCD

I, C

1. Bisect z. A and D. Let bisectors meet at G. 2. Measure the following:

mDAG= mAGD= mLGDA=_________

3. Bisectz B and LC. Let bisectors meet at H. Draw GH.

4. Measure the following:

mLCBH= mL.BHC=____________

m.HCB=________

5. Extend GH so that it intersects AD and E and CB and F.

6. Measure AD, AE, ED to the nearest 1/2 cm.

Measure to the nearest Y2 cm.

7. Measure bt, , A8 to the nearest 1/2 cm.

8. Study the constructions you have made and write down as many conjectures (tentative conclusions) as you can.

9. Prove or disprove your conjectures.

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LESSON PLAN NO. 4 GEOMETRY & ALGEBRA - Separation Points for Line Segments

1. Graph the line segments that have the following number pairs as end points:

a) (6, 3) , (10,9) d) (-7, 5) , (-7, —3) b) (-5, 6) , (+7, —2) e) (-12, 3), (-2, —9) c) (-3, —8), (5, —6) f) (6, —3) , (-8, —3)

2. Locate the mid-point on each of the line segments above and give the co-ordinates (or number pair) that describe the point.

3. Could you have determined the mid-point without graphing the line segments? If you think you can, predict the mid-point for segments having the end points. a) (6, —2) , (-4, 8) b) (4, —7) , (-5, 2) Check your predictions by plotting the points and measuring.

4. Assume the end points of a line segment are (x 1 ,y 1 ) and (x2,y2). What would

be the co-ordinates of the mid-point of this segment?

5. From what you think is true about the co-ordinates of the mid-point of a line segment, make conjectures concerning the co-ordinates of a point one-third the distance from points P to Q below.

P Q Conjectures Correct

(0,0) (12,6) (-3,3) (6,-9) (-8,-5) (4, 1)

6. Graph the points and draw the required segments. Check your conjectures to see if they are correct. If they are not, record the co-ordinates of the correct point for each segment on the appropriate blank in Exercise 5 and then revise your scheme for obtaining the co-ordinates so that you do get the correct result. Try your revised scheme (or your original scheme if your conjectures in 5 were correct) on the points below. Be certain to check your results by the graphing technique.

P Q P Q

a) (7,-8) (10,10) c) (11,2) (10,-7)

b) (-15, 10) (9,-5) d) (-10, —8) (-10,7)

7. Assume that the end points of a line segment are (x 1 ,y 1 ) and (x 2,y2). What would be the co-ordinates of a point 1/3 the way from the first to the second point. Use the co-ordinates of the points in Exercise 6 to check your formula.

8. Revise the formula in Exercise 7 so that it can be used for finding the mid-points of the segments in Exercise 1. Your revisions should be as few as possible.

9. Revise your formula so that it will hold for finding any point Fa of the distance

from one point to a second point. Check your formula empirically by trying

a few cases. Prove your formula is true for any pair of points where is any

rational number.

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BCTF Lesson Aids The following is reprinted from the latest BCTF Lesson Aids Catalog. Send all orders to BCTF Lesson Aids, 105 - 2235 Burrard Street, Vancouver, B.C. V6J 3H9. Please be sure to enclose amount (minimum order 50d) name, address and school district no.

NO. ARITHMETIC - MATHEMATICS Price 8402 ARITHMETIC EXPERIENCE CARDS, 26 P. 1971. Drawings. A set .60

of suggested assignment cards providing ideas for a prac-tical approach to teaching measurement at the primary grade level. Should be trans ferred to filing cards.

8403 IDEAS FOR ARITHMETIC GAMES by Gail Sear. 5 p. 1972. .12 Drawings. Describes games on such things as plus and minus quantities, the symbols for greater and lesser, addition, and multiplication.

8429 NUMBER RECOGNITION by Janeine Gentile, 2 p. .60 1973. Drawings. A laminated device for chil-dren to match quantities to the numerals 2 to 9. Based on the jigsaw principle. 2

The following six games are laminated, self-checking jig-saw puzzles designed to give practice in arithmetic funda-mentals. They are based on an idea submitted by Sheila Page.

8422 HIPPOPOTAMUS PUZZLE, 2 p. 1972. Addition. .60 8423 FISH PUZZLE, 2 p. 1972. Subtraction. .60 8424 SNAKE PUZZLE, 2 p. 1972. Multiplication tables 2 to 5. .60 8425 BUTTERFLY PUZZLE, 2 p. 1972. Division by 2 to 5. .60 8426 FROG PUZZLE, 2 p. 1972. Multiplication tables 6 to 9. .60 8427 DUCK PUZZLE, • 2 p.-1972. Division by 6 to 9. .60

8430 HELP THE ANIMALS by Janeine Gentile, 2 p. 1973. Drawings. .60 A laminated device for giving practice in simple arithme-tic fundamentals. Child matches problem cards to answers.

8431 ARITHMETIC DEVICES by Janeine Gentile and Valerie .10 Bortolleto, 4 p. 1973. Drawings. Some tips and sugges- tions, and a series of devices to make. Primary.

8428 PLACE VALUE by Janeine Gentile, 1 p. 1973. A lamiCated .30 sheet which, when cut up and stapled, forms four pockets plus 38 numbered strips for giving practice in making and reading numbers up to four digits.

The following four items are laminated activity cards approximately 4" x 4". All are for Grades 2-3.

8432 ACTIVITY CARDS FOR AN ABACUS by Maimee Tomlinson, 1 p. .30 1973. Six cards.

8433 ACTIVITY CARDS FOR ATTRIBUTE BLOCKS by Maimee Tomlinson, .60 2 p. 1973. Twelve cards.

17

NO. ARITHMETIC-MATHEMATICS - (continued) PRICE

8434 ACTIVITY CARDS FOR COUNTERS by Maimee Tomlinson, 1 p. .30

1973. Six cards.

8435 ACTIVITY CARDS FOR GEOBOARDS by Valerie Bortolleto and .60

Maimee Tomlinson, 2 p. 1973. Twelve cards. Activities

are for 1" regular boards.

8436 DISCOVERY BOARD AND ACTIVITY CARDS by Maimee Tomlinson, 1.20

4 p. A laminated card marked off into 100 squares with accompanying strips representing 10 and squares represent-ing one. Laminated activity cards are provided. Used for individual practice in grouping 10's and l's.

8404 POLYHEDRAN BALL, 2 p. 1973. Each face of this laminated .60 polyhedran has a multiplication question. Children can create pair and circle games.

8405 ADDITION AND SUBTRACTION FLASH CARDS, ..... 1.80

6 p. 1973. Devices for instant recognition of addition and subtraction facts to 10. Strips of laminated cardboard with colored geometric shapes. No numerals.

8406 NUMERATION FLASH CARDS, 9 p. 1973. One inch circles 2.70 representing the numbers from 2 to 10 arranged in both random and domino patterns. Laminated.

8407 ADDITION JOB CARDS by Maimee Tomlinson, 12 p. 1973. 1.80

Three sets of laminated job cards (1) identifying 10's and

l's in 2-digit numbers, (2) adding two numbers giving the

answer in 10's and l's, (3) adding a 2-digit and a 1-

digit number with carrying. Self-checking. About 4"

x 4".

9402 C & G MASTERY CARD SET by R. T. Cuthbertson and W. L. 1.00

Ginther, 9 p. 1964. A method of facilitating the mastery

of fundamentals. Consists of seven cards, containing work

at various levels of difficulty. Comes with directions and a sample mastery card graph for recording pupils'

progress. An efficient way to reduce time spent on arith-

metic drills.

9403 MATH RUMMY by Gail Sear, 3 p. 1972. 52 laminated cards, 1.00

each containing a simple mathematical problem (addition,

subtraction, multiplication, or division). The answers to

these problems make four sets of numbers from 1 to 13. Each of the four sets is designated as a suit through

different geometric shapes. Children use them to play

rummy. Suitable for ages 10-12.

18

NO. ARITHMETIC-MATHEMATICS - (continued) PRICE

9406 THE METRIC SYSTEM by C. V. Johnston, 11 P. 1973. Twenty- 1.80 one laminated activity cards approximately 4" x 4". By working through these cards, students develop an under-standing of linear, weight and volume metric units. Suit-able for ages 9-12.

9404 BASIC FACTS IN ARITHMETIC by L. R. Mawhinney, 2 p. 1968. .05 Describes a technique for reinforcement.

9405 SUPERMARKET MEASUREMENT by David Weller, 5 p. 1972. A .12 series of questions for investigating volume, weight, mass and unit value. Empty, labeled containers are used in the study.

9412 REVIEW OF FRACTIONS, 1 p. 1964. A 50-question test or .02 exercise.

9413 NEW MATHEMATICS WITH THE PIONEER'S TALLY by C. A. Hensley, .12 5 p. 1966. Drawings. Tallies provide manual as well as visual impressions in counting, adding and subtracting. They immediately clarify the concept of numbers in bases other than 10. The article contains patterns and full directions for making and using tallies. Teacher refer-ence.

9415 AID FOR DRILL IN DECIMALS AND FRACTIONS, 6 p. 1964. .12 Teacher reference. Describes a method of giving needed practice.

9419 FOUR PERCENTAGE TESTS, 2 p. 1964. Each test has ten .04 questions and five problems. Includes a key. Should be put on filing cards.

9424 ELEMENTARY PLANE GEOMETRY by J. L. Ferrari, 12 p. 1964. .20 A booklet suitable for students as a reference or work-book. Some terminology must be changed to suit the current course.

3003 AREA AND PERIMETER by W. D. Finley, 8 p. 1971. Drawings. .20 Some geometric figures suitable for practice in measuring and in computing area and perimeter. Elementary students could work with some of these.

3004 GEODESIC MODELS by Dave Gemmell, 14 p. 1973. Drawings. .35 Teacher's manual outlining projects which lead to an understanding of geodesic domes. Students must have a basic knowledge of arithmetic and algebra.

19

NO. ARITHMETIC-MATHEMATICS - (continued) PRICE

3027 FACTORING by R. E. Fleming, 34 p. 1967. A supplementary .75 unit emphasizing the use of the distributive principle. Includes the application of factoring to the solution of quadratic equations. Many exercises with answers pro-vided. Student and teacher use.

3026 MATHEMATICS FOR PHYSICS AND CHEMISTRY STUDENTS, 79 p. 1.00 1967. A booklet designed by physics and chemistry teachers to help students having trouble with science because of mathematics. Arranged so that students can work through the procedures with a minimum of teacher help. Compiled by science teachers under the auspices of the B. C. Science Teachers' Association. (Cross-refer-enced to Science.)

3028 BINOMIAL EXPANSIONS by C. Homer, 20 p. 1971. Can be .50 used from Grade 9 up, but is particularly applicable to Math 12. May be used either to introduce the topic or for remedial work.

After the topic has been introduced, a discovery approach is used to compare the coefficients of terms in a binomial expansion with Pascal's Triangle. The same method is then employed to 'discover' other patterns in the expansion. Later, both a recursive formula, and a general formula for any term are-presented-and their uses illustrated.

The programmed instruction format suits the topic and makes it easy for students to follow the step-by-step development.

1234V678 THERE'S A FOREIGNER AMONG US

Football Poker by S.G. Bell

OBJECTIVE The student adds directed numbers.

MATERIALS 1 card deck •) 1 playing field per student pair. 1 paper clip )

Football poker is a card game that utilizes the gains and losses aspect of football to introduce addition of directed numbers. Gains are represented by black cards, losses by red cards. Students find the net gain or loss resulting from a series of cards or 'downs' taken together. The game is played on a diagram of a football field, with a paper clip or some other device being used to mark the position of the ball (see Figure I).

FIGURE I It is convenient to use a paper clip bent as below to indicate the position OT the ball.

The game involves two players, one taking the part of a CFL team, the other the part of an NFL team. Since Canadian and American rules differ, new rules must be decided upon. A compromise is reached by having the first quarter played under the CFL rule of three downs to make ten yards, the second quarter under the NFL's four down rules, and the second half under a five down rule. The Americans are so confident, they agree to play on a CFL size field.

Rules will be discussed only for the second half of the game, on which there are five downs. They can easily be inferred for the first and second quarters, the only difference lying in the number of downs. Only four of the five downs may be used to make the ten yards re-quired for a first down. The fifth must be used to punt. On a punt

21

the ball travels 30 yards and there is a change in possession. The player losing possession shuffles the deck and deals four downs to his opponent. If the ball lands in the end zone on a punt, a score of one point is made and the oppon-ent receives possession on his 25 yard line. A player dealt four downs finds the net gain or loss resulting and moves the football accordingly. If there is a net gain of ten or more yards, the player receives another hand of four downs from the top of the deck. Otherwise, the player must use his/her fifth down and punt. A touchdown is made when the ball is advanced into the end zone and is worth seven points. A kick-off occurs either after a touchdown, or at the beginning of the game, or at the beginning of the second half. Who re-ceives at the beginning of the game is decided by a draw from the deck - high card wins. The player not receiving at the beginning of the game re-ceives at the beginning of the second half. A player receiving a kick-off is given the ball on his 30 yard line and dealt a hand of four downs. Note that since the wind factor is negligible, the teams have decided not to bother changing ends at the quarter. To add interest to the game, a black face card counts as a pass completion of 15 yards, while a red face card counts as a fumble and results in immediate loss of the ball. The jokers may be added to the deck and given special designations, such as a touchdown run or an inter-ception. Also, if a player notices his/her opponent make an error in finding the net gain or loss, the player is entitled to charge the opponent a penalty in which any gain in the play is wiped out and a 10 yard loss substituted. In such a situation the opponent retains possession and is given first down over again, but this time has 20 yards to go. He/she receives four new cards from

the top of the deck.

A possible follow-up to the game as outlined above would be to have a second game, but this time use a deck of 2Y2 1 ' by 3h/2' cards with directed numbers printed on them. These cards could include directed numbers of large absolute value. Such a follow-up would facilitate the transition from gains and losses to positive and negative numbers.

After the students have played the game they should be able to add series of directed numbers. The verbally-solved problems of the game should then be related to written problems. At this time, a large number line on the wall facilitates presentation of an alternative solution method to that of losses and gains - trips on the number line. A student volunteer is obtained to take the part of the quarterback of a football team. He/she begins at zero, or the line of scrimmage. His/her path is then charted through successive losses due to fictitious towering linemen and gains re-sulting from his/her own bravery and prowess. Hence, at the end of the lesson the student has two methods for solving problems in addition of directed numbers. He/she also has achieved the more important goal - proving once and for all that the CFL is superior to the NFL.

22

FOOTBALL POKER RULES

I Card Values

Each card counts as one down. 1. Black card (other than a face card) - gain equal to the value of the card. 2. Red card (other than a face card) - loss equal to the value of the card.3. Black face card - pass completion of 15 yards. 4. Red face card - fumble and immediate loss of the ball.

II Kick-Offs

Player receiving the kick-off is: 1. given the ball on his/her 30 yard line. 2. dealt a hand of downs.

Ill Punts

1. The ball travels 30 yards. 2. Possession changes. 3. The receiver is dealt a hand of downs.

IV Advancement of the Ball

1. The ball is moved according to the net gain or loss in the hand. 2. If there is a net gain of 10 or more yards, possession is retained and a new hand of downs received. 3. If there is a net gain of less than 10 yards or a net loss, the player must punt.

V Scoring

1. If the ball lands in the end zone on a punt, a score of one point results. 2. If the ball is otherwise advanced into the end zone, a touchdown or seven point score is made.

VI Penalties

If a player notices his/her opponent make a mistake in finding the net gain or loss, the player can charge the opponent with a 10 yard penalty. On a penalty: 1. possession is retained. 2. first down is given over again, but this time with 20 yards to go. Four new cards are dealt from the top of the deck.

23

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POKER 24

Book Reviews by Roger Sandford

Mathematics, A Human Endeavor: A Textbook For Those Who Think They Don't Like the Subject, by Harold A. Jacobs. W.K. Freeman & Co., San Fransisco, 1970, 529 pages.

This book was reviewed by Mike Baker some time ago, and his rave review made me get it. It's one hell of a good book! It was so good I managed to persuade my principal to buy a classroom set - I am a rotten persuader - the book sold itself to a nonmathematician! It is tailor-made for the literate arts student, but I used it with three GM 11 classes, using it 50% of the time. They were not too literate, but by my reading the exposition with the students and explaining the occasional polysyllable, the students were introduced to aesthetically pleasing, amusing, interesting and varied topics, all of which lead to real, but not heavy, mathematics. This is the first textbook I have ever used in a classroom that leaves me speechless with admiration. Why not get a copy (perhaps on approval) and see whether your reviewer is a starry-eyed fool, in the pay of the publisher or Mr. Jacobs, or, in fact, dead right in saying this is the most worth-while textbook written in the last x number of years (N.B. it's written by a practicing classroom teacher in a JH School).

by P. Peak

The Slow Learner in Mathematics. Thirty-fifth Yearbook of the National Council of Teachers of Mathematics. William C. Lowry, ed. 1972, xiv + 528 pages. Available from the National Council of Teachers of Mathema-tics, 1906 Association Drive, Reston, Virginia 22091. $8.50 postpaid ($7.50 to members of the NCTM)

As indicated in the title, this yearbook is designed specifically to help the teacher of the slow learner of mathematics teach better. However, there is considerable material in the book that is highly beneficial to all teachers of mathematics. It goes beyond the processes to be used to help students reach specific cognitive mathematics objectives by including those that also help the learner improve his self-image, his confidence in his mathe-matics ability, his desire to learn more, his skill in using what he knows, and, most of all, his satisfactions while carrying out the processes of learn-ing mathematics. The theory included is used to clarify the rationale for the process. The reader will obtain helpful information about most situa-

25

tions in which he/she may be called upon as a teacher. The yearbook committee has wisely refrained from trying to precisely define the slow learner but takes the approach that a reader will find help from this text in teaching those he/she has individually defined as slow learners in math-ematics.

The book presents what research has found about the learner, the techni-ques for selecting appropriate objectives, the kinds of learning environments that are necessary, and background material to help put the characteristics of the slow learner of mathematics in proper perspective. There are five chapters that present specific helps for teachers through descriptions of multisensory aids, teaching styles at both elementary and secondary levels, mathematics laboratories and the activities within them, and diagnostic-prescriptive teaching which is so essential if we are to assure student pro-gress. One chapter, on 'promising practices,' provides both the preservice and in-service teacher an excellent opportunity for careful analysis as to which practice will fulfill its promise. Some attention is given to admini-stration and to training of teachers. The appendix contains 42 pages of games, applications, bibliography, and eight sample lessons.

Each teacher should have one available for ready-reference when he/she has the feeling of frustration that comes from failing to reach his/her pre-

determined goals.

NCTM MEMBERSHIP AS OF MAY 31, 1973 1973 1972

Canadian 31075 2,815 + 260

Alta. 337 337 0

B.C. 393 401 - 8

Man. 172 215 - 43

N.B. 177 136 + 41

Nfld. 73 43 + 30

N.W.T. 1 1 0

N.S. 198 82 + 116

Ont. 1,249 1,232 + 17

P.E.I. 1 1 0

Que. 319 285 + 34

Sask. 155 82 + 73

26

The Diagnostic Prescri ption Clinic FOR EXCEPTIONAL LEARNERS IN MATHEMATICS sponsored by the University of British Columbia's Mathematics Education Department.

Many, perhaps too many, school children seem out of place with the standard mathematics curriculum. Many of these youngsters can be helped by receiving extra attention from their mathematics teachers, while others need more specialized attention. The Mathematics Education De-partment of the University of British Columbia operates a clinic that will provide, free of charge, a diagnosis of a student's mathematical knowledge and a detailed prescription of an appropriate mathematics program for that student. The clinic is designed to assist students in the following categories:

- students who are capable of advancing very rapidly in mathematics.

- students who have fallen significantly behind due to physical or psychological problems.

- secondary students who are having unusual problems with a given mathematics course.

- students who have transferred from a different school system and seem to have large 'gaps' in their mathematical knowledge.

The Mathematics Education Clinic can provide detailed diagnosis and prescriptions ONLY. Implementation of the prescriptive program must be carried out by a competent tutor selected by the professional staff refer-ring the student to the clinic. The tutor will be asked to supply weekly written reports on the youngster's progress to the Mathematics Education Clinic. Such reports are the feedback necessary so that the initial diagno-sis and prescription can be adjusted as the student's progress indicates.

Because the demand for a service is so great and because of the practical and financial limitations of this initial effort for the clinic, it will be necessary to attempt to select those youngsters who will most benefit from the service of the clinic. For these reasons, it is suggested that school referrals come from school counsellors or principals. Interested parents and teachers should contact such people if they know of a youngster in need of the clinic's service.

27

For further information please communicate with:

Dr. David Robitailie Faculty of Education University of British Columbia Phone: 228-5337 or 228-2141 (if no answer)

New Books across My Desk 1. Mathsense One by Douglas Allies; Ginn and Company, Toronto,

1973. $2.50 (Teacher's Guide 800 The first of a two-book series of mathematics in a game setting. Mathsense is a series of games and puzzles designed to create interest and attract attention. A good supplementary aid to reinforce com-prehension of basic arithmetic skills. Of special interest at the inter-mediate grade levels.

2. Statistics By Example - prepared by a joint committee of the NCTM and the America Statistical Association, the four-volume series is available from Addison-Wesley, Sand Hill Road, Menlo Park, Calif. 94025. The cost is $2.25 per volume and $1.50 for each Teacher's Manual per volume.

3. A selected bibliography of instructional aids for metrication prepared by a joint American Association of School Librarians (AASL) and NCTM committee is available for 2(k from the AASL, 50 East Huron Street, Chicago, Illinois 60611.

3 1 I was a B until my accident. A meeting of the lines club.

28 PsA73-147/hk