1976 – 17(3)

BRITISH COLUMBIA ASSOCIATION OF MATHEMATICS TEACHERS
NEWSLETTER/JOURNAL
VOLUME 17, NUMBER 3 MAY 1976
BCAMT EXECUTIVE 1975-76
PAST PRESIDENT
Alan Taylor
7063 Jubilee Street
Burnaby, B.C. V5J 4B4
434-6315 (home)
936-7205 (school)
1st VICE-PRESIDENT
Roger Sandford
R.R.1, Tzouhalem Road
Duncan, B.C. V9L 1M3
746-6418 (home)
746-4434 (school)
TREASURER
William A. Dale
1150- 17th Street
Courtenay, B.C. V9N 1Z7
338-5159 (home)
CURRICULUM CONSULTANT
William J. Kokoskin
1341 Appin Road
North Vancouver, B.C. V7J 2T4
988-2653 (home)
988-3161 (school)
MEMBERSHIP SECRETARY
Marion Harvey
946 Gatensbury Street
Coquitlam, B.C. V3J 5J3
939-5488 (home)
939-4201 (school)
ELEMENTARY REPRESENTATIVE
Grace Dilley
2210 Dauphin Place
Burnaby, B.C. V5B 4G9
299-9680 (home)
596-0357 (school)
PRESIDENT
John Epp
1612 Wilmot Place
Victoria, B.C. V8R 5S4
592-2388 (home)
478-5548 (school)
RECORDING SECRETARY Ian C. deGroot
3852 Calder Avenue
North Vancouver, B.C. V7N 3S3 980-6877 (home)
987-7178 (school)
NCTM REPRESENTATIVE
Tom Howitz
Faculty of Education, UBC
2075 Wesbrook Place
325-0692 (home)
228-5203 (UBC)
PUBLICATIONS CHAIRPERSON
Susan J. Haberger
1390 Willow Way
Coquitlam, B.C. V3J 5M3
939-8618 (home)
936-7205 (school)
SUMMER WORKSHOP 1976 Doug Owens
Faculty of Education, UBC
2075 Wesbrook Place
596-0718 (home)
228-4808 (UBC)
IN-SERVICE SPECIALIST
Dennis Hamaguchi 3807 22nd Avenue
Vernon, B.C. V1T 1H7 542-8698 (home)
542-3361 (school)
NORTHWEST NCTM CONFERENCE ORGANIZER Trevor Calkins
1623 Amphion Street Victoria, B.C. V8R 4Z5 592-4463 (home)
592-1205 (school)
Inside This Issue
5
President's Message .................................... John Epp
7
Your Executive Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sue Haberger
GEOMETRY - BOTH SIDES NOW
8
10
An Open Letter to Ken Silen.........................Richard Longman
Letter to the Editor
10
3rd International Congress of Mathematical Education
11
18
Transformations and the New Senior Geometry Course ......... Walter Szetela
Deductive Geometry in the Junior-Secondary Curriculum . ......... Robert Peard
25
The Brick-Laying Problem.............................. Pat Presidente
PRIMARY MATHEMATICS TEACHING
28
The Hundred Square ................................. Olive Stewart
33
The Child's World: Mathematics ........................ .Werner Liedtke
TEACHING GRADE 9 AND 10 MATH
38
Using 'Mathematics for a Modern World' ...................... Wilf Baxter
LESSON PLANS
Addition Shade In and The Five Square Puzzle ................ Grace Dilley
Tangrams, Find a Match and
Addition-Subtraction-Multiplication-Division ............. Dennis Hamaguchi
INTERMEDIATE MATHEMATICS TEACHING
50
A Unit on Numeration Systems ........................... Peter Makeiv
57
Math Games for Intermediate Grades .................. R. Melendez-Duke
62
Learning Difficulties in Mathematics. . .................... David Robitaille
66
Election Results
67
Metric Measure .................................... Jane Srivastava
COMPUTING SCIENCE
71
Computer Science at Churchill Secondary ................ James Nakamoto
75
More Problems for Computer Science Students................. Doug Inglis
78
Minicalculators in Our Schools 1975..................... Joseph Caravel la
79
Calculator Games .................................... John Petrak
82
List of Publisher's Representatives ....................... Bill Kokoskin
85
Fifth Mathematics Summer Workshop...................... Doug Owens
86
Northwest Conference ............................... Trevor Calkins
VECTOR
VOLUME 17, NUMBER 3 MAY 1976
3
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 1976-76 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
4
President's Message
Dear Colleagues:
-
As school ends and you are looking forward to a well-deserved rest over the summer,
remember the Fifth Mathematics Summer Workshop on August 31 and September 1 at
Carson Graham Secondary School in North Vancouver.
Attending this would be an ideal way to shift your mind back into gear for the fall session. Don't forget the AGM will also be held during this time at the same location, and
this year, will not conflict with either your lunches or with the program.
My personal feelings are that one of the most important functions of the BCAMT is to
provide leadership in in-service. In the coming year we have two excellent programs arranged. The summer workshop once again shows great promise, and the 1976 Northwest
Math Conference on October 29 and 30 at the Empress Hotel in Victoria has very many
top national and local speakers. At both of these, the speakers cover such a spectrum of
topics that it should provide a terrific opportunity for both elementary and secondary
math teachers to learn new ideas, become revitalized and eager to return to the classroom.
Both programs should provide not just something for everyone, but an abundance for
everyone.
As you may have noted, I become rather enthusiastic when I see the programs. I hope
you share my enthusiasm and are able to attend at least one.
There are other in-service activities which are also valuable. These are the ongoing inservice activities that occur in the individual locals and on the Professional Day.
Some people tend to think of Professional Day as a waste of time but in my district the
math teachers prepare the math program for the day and each of the other areas organize their own programs for the day. If careful thought and preparation have been given,
the day is a success
Professional Day and other in-service activities are opportunities to exchange ideas and
information informally and to listen to speakers present ideas and topics.
If you can teach in a classroom, you obviously have some pet techniques. Share these
ideas so others can try them or adapt them to suit their own classrooms.
Here's one idea to start you off. When my students are hard to motivate (for whatever
reasons), I give them a Quickie Quiz. It is a short quiz of four to five questions that
relates to the work taught in the previous class. I allow the students five to ten minutes
to finish the quiz. The marks, five to ten per day, are accumulated and students see
immediate results from their work. A student must be very blasé not to make an effort
to try to get some of these easy marks, and therefore puts in some time each night on
math.
So get together and share ideas. Remember an old idea to you may be new to someone
else. I hope to see you all at the summer workshop and at the northwest conference.
Respectfully yours, John C. Epp
Your Executive Committee
SUE (ROWLEY) HABERGER
Sue, acting publications editor for the BCAMT, has now been elected to fill
the position for the next two years. She graduated from UBC in 1968 and
traveled to Sydney, Australia where she obtained an M.Sc. degree in Mathematics. As a graduate student, she did a considerable amount of tutoring,
and the experience led her to enroll in the PDP program at Simon Fraser
University when she returned to Vancouver in 1973.
Since obtaining a teaching certificate, Sue has taught one year at Burnaby
North Senior Secondary School and is currently teaching at Centennial
Senior Secondary School in Coquitlam.
She finds the job of putting Vector together demanding but enjoyable, and
she welcomes letters, articles, or short lesson plans from B.C. teachers at all
grade levels. Vector is a publication designed for mathematics teachers to
share ideas and to communicate with one another, and Sue's job is to maintain the publication's high quality.
7
GEOMETRY - BOTH SIDES NOW
An Open Letter to Ken Silen
FROM: Richard S. Longman
George Pringle Secondary School
Westbank, B.C.
January 13, 1976
Dear Mr. S ilen
Your recent letter to the publishers of Jacobs' Geometry, printed in the October
1975 Vector, has led to some further examination of that text.
There are many more good features than your letter specifically mentioned, but
there are some aspects that seem in need of improvement.
A table of symbols would be useful.
The index appears to have been written mainly from the lists of 'Basic Ideas' found
at the ends of the chapters. This might have been less expensive than some other methods,
but some terms seem to have been omitted (e.g., 'proof) and others seem to have been
misplaced (e.g., 'length of line segment' is listed for page 77, and it should have been listed
for page 73, in that 'AB' is explained there).
Why use ' r and 'L' for 'angle' and 'measure of an angle,' respectively? Are these in
current use by others? Do not 'L' and 'mL' seem less confusing?
Why is the phrase 'equal angles' used to convey the idea of-'equal-measured angles'?
The frequent use of 'equal sides' instead of 'equal-length sides' is similarly questioned.
The introduction to the handling of proportions seems rather good, but with the new
B.C. geometry course's being made a senior elective, this presentation would be a bit late
for most students.
The explanations of 'converse,' 'inverse' -and 'contrapositive' ('converse-inverse' for
the British types) on pages 18 and 19 are appreciated and have been sorely missed in the
recently used texts.
The definition of 'corollary' on page 93 is much better than that found in the current
Ma 10 text.
The explanations and use of 'if' is greatly liked, especially in contrast to the material
found on page 39 of the presently prescribed Ma 10 text.
The use of simple flow charts to explain conditional statements is fine, but why has
no further use been made of them to help explain any of the proofs? How many students
have become confused because of thinking that every proof follows a linear pattern?
(E.g., the six-step proof of page 123 follows a pattern such as:
(1)—'(2)
(5)
8
(6)
and problem 2 of page 175:
(a)(b)
(d)
(c)
(e)—+(h)--(j)--(k)
(f) -' (g)--"
[The above owe quite a bit to Frank Allen, former NCTM president and his presentation
at the August 1966 Calgary NCTM meeting.])
Your reference to the spirit-master tests brought a feeling of disquietude. I have not found a series of publishers' tests to my complete liking. Is the use of true-false (or other obvious two-choice) items avoided? Is there provision made for students to display reasoning ability without being unduly restricted by the format of the test? If multiplechoice items are used, is the humor displayed in the book evident in the distractors presented? Are the tests designed to teach, as well as to measure student programs? Most important, are tests to be revised and republished at least annually so that teachers feel free to return corrected tests to students? (Some teachers have been known to permit their students to keep no copies of tests, to resort to locked file cabinets, and to give the same tests year after year. For some, the use of published tests seems to encourage such practices, which I put on a par with the repeated use of ancient, yellowed, lecture notes.) This letter could go on to much greater length but seems rather long at this point. Of the twenty-odd geometry texts which I keep for the use of my students, the Jacobs text is one of the most-used; It is hoped that it will be even better in future editions.
Sincerely,
Richard S. Longman
Mathematics Teacher
cc: Mr. Kudlacik, Freeman and Co.
Editor, Vector
9.
Letter to the Editor
April 2, 1976
Dear Mrs. Haberger:
The article, 'Computing Curriculum - Preliminary Draft' (Vol. 7, No. 2) submitted
by Bill Kokoskin fails to mention that the outline was not entirely the work of the Math
Revision Committee. To put it precisely, it was prepared by the Curriculum Subcommittee of the B.C. Computing in Education Committee, for adoption by the Math Revision
Committee.
As a member of the above subcommittee, I assure the information I am presenting
you is quite correct.
Sincerely yours,
James Nakamoto
Math/Computer Science Department
Churchill School, 7055 Heather Street, Vancouver, B.C. V6P 3P7
3rd INTERNATIONAL CONGRESS
ON MATHEMATICAL EDUCATION
KARLSRUHE - AUGUST 16 - 21, 1976
The organizing committee has published a second announcement giving details of sessions and speakers.
Interested persons are asked to contact the committee
directly to obtain copies, registration forms, and accommodation information.
The Local Organising Committee
in Karlsruhe
Professor Dr. H. Kunle
Secretary: (E. F. an Huef +) J. Mohrhardt
Postal Address
3rd International Congress on Mathematical
Education 1976
KaiserstraBe 12, University of Karlsruhe
D 7500 Karlsruhe
(Federal Republic of Germany)
10
Trans f ormations and the New
Senior Geometry Course_______
by WALTER SZETELA, Department of Education, UBC
When Dieudonne made his famous statement in 1961, 'The present year-long course in
Euclidean geometry must go,' he stirred smouldering embers that ignited extensive and
continuing controversy. There was substantial agreement that, although proving theorems
was a stimulating experience for many students in secondary school mathematics, the
course failed to fit the needs of most students, that geometry, as taught, failed to mesh
with other branches of mathematics, and that the content of geometry textbooks was
little different from 19th century textbooks. But the big question remained. If the standard geometry course must go, what should replace it?
The geometry textbook by Moise and Downs was a result of recommendations by SMSG
in the early '60s. Major changes included the following:
1. Certain loopholes in the Euclidean axiom system were plugged with the help of such
properties as order, separation and incidence, and by grounding geometry in the, realnumber system. (Greater rigor often brought greater confusion.)
2. Language was made more precise, and definitions were more carefully presented.
(Unfortunately the attempts at such care and precision made statement more correct
but less intelligible.)
3. Solid geometry was added to the course content. (Often this resulted in more frustration by students unable to wrestle with theorems and originals in flatland.)
4. Some co-ordinate geometry was included in the content. (This was, perhaps, the
most useful change or addition - one most teachers agree should be included in the
geometry course.)
NOTE: In a 1973 nation-wide survey of 605 secondary mathematics teachers in the
United States, 85 percent of the teachers felt that co-ordinate geometry should be included in the course (Gearhart, 1975).
Meanwhile, in British Columbia, a great deal of intuitive geometry was being introduced
in Grades 1-9. A brief glance at the curriculum guide will make this strikingly obvious. In
view of the circumstances, the provincial curriculum revision committee decided to abandon the Grade 10 standard geometry course and to replace it with a Grade 9 and 10 math
program that included a combination of arithmetic, algebra and geometry. Nearly all the
geometry was in the nature of measurement and computation, with some geometric constructions, but no deductive geometry. Many teachers in British Columbia were unwilling
to give up the deductive geometry course, as witness the number of schools still using the
Moise and Downs text in Grade 10. The revision committee decided that a deductive geometry course should be presented as a Geometry 12 course that could be elected in either
Grade 11 or 12. In recognition of the inadequacies of the traditional deductive geometry
11
course, the revision committee decided that the geometry course should include many
approaches, should include a wide variety of problems, should build on the intuitive geometry concepts developed in Grades 1-10, should exhibit the relationships of geometry,
algebra, and arithmetic, and should still provide opportunities for students to prove theorems and originals using axioms and postulates. However, deduction and proof are introduced rather late in the course to insure that students have considerable skills in coordinate geometry and in concepts of geometric transformations. Indeed, the study and
applications of geometric transformations are by far the most significant changes in emphasis in the proposed geometry course. To many secondary school teachers, the notion
of geometric transformations is somewhat vague, and there is a tendency to resist attempts to teach a significant amount of geometry using transformations. In the Gearhart
survey cited earlier, only 47 percent of teachers agreed that geometric transformations
should be included in the course. Only 11 percent agreed that the course should be based
on transformations. However, in both cases, about 30 percent of the teachers were undecided about the use of transformations. This may reflect a lack of knowledge about geometric transformations rather than any antipathy for transformations. In fact the concepts of these transformations, namely the line reflection, rotation, and translation are
curriculum objectives for Grades 4, 5 and 6, so that such transformations may alread.y be
familiar to students taking the senior geometry course. There is also a substantial amount
of geometric transformation in the Mathematical Pursuits textbook series. While it must
be made clear that the new geometry course presents several geometries, including the
familiar co-ordinate geometry, real-world geometry, vector geometry and transformation
geometry, this paper is intended to try to remove some of the mystery from transformation geometry and to illustrate that it may serve equally as well as traditional geometry
as a vehicle for proving theorems. In addition, it serves to clarify the study of vector geometry, which is basically geometric transformations called translations. Further more it
makes easier the proof of some theorems that would be difficult with traditional geometry.
Rather than define the various transformations, the following diagrams should provide
sufficient understanding of the concepts to understand the theorems that will be used as
examples to demonstrate that transformations can be used just as effectively as traditional methods in making proofs.
The Line Reflection
tn
Figure 1
-
Figure 1 shows a reflection of the letter F in the line m. Note the following:
1. The reflection is a congruence transformation. It preserves distances and angles.
2. The line of reflection is the perpendicular bisector of the line that joins a point with
its image after reflection in m.
3. Reflection changes the orientation of a figure.
12
The Rotation
MMMEMMOREMIMMI
MMEREMIMMUMMM
MMINIMEMINIMIMEME
Figure 2
Figure 2 shows a rotation of triangle ABC about 0, the center of rotation, through 90
degrees. Note the following:
1. The rotation is a congruence transformation preserving distances and angles.
2. The angle made by a point, the center of rotation, and the image point is the angle
of rotation.
3. Rotation preserves the orientation of a figure.
The Half Turn
maariiaasaau
MMMMWLMMMMMMMM
NOMWEEMEMI NONE
auiaiIriaIua
4 00
M E 01.10
ENIONEIMM
momME MENNEEME
M
Figure 3
Figure 3 shows a half turn of line segments about 0, the center of rotation. Note the
following:
1. The half turn is a congruence transformation preserving distances and angles.
2. The center of rotation is the bisector of the line segment joining a point and its
image after the half turn.
3. The half turn preserves orientation.
4. The half turn is a special rotation with angle of rotation 180 degrees.
13
MENNEMEMOMMMUM
F'
The Translation
MMENSMEMEMERFM
MMMPAFAMEMEMPAPEEMI
MERENEENEEMMEN
INEEMMENNEENEEN
Figure
Figure 4 shows a translation, a vector transformation. Note the following:
1. The translation is a congruence transformation.
2. Under translation lines go into parallel lines.
3. Translation preserves orientation.
4. This particular translation could be represented by the vector (8
The first example of a proof using transformations appears in the geometry textbook
recommended by the revision committee, written by Graening and Nibbelink, on page
326. The traditional proof appears as a problem in Moise and Downs on page 265.
P
R
The proof makes use of a generalization discovered earlier: A quadrilateral is .a parallelogram if and only if it is conserved under a half turn. The proof will proceed by showing
that the quadrilateral AQBS will be transformed into its identical outline, BSAQ, after a
half turn about 0, the midpoint of diagonal PR.
THEOREM If PORS is a parallelogram and if A and B are points on diagonal PR such
that PA
RB, then AQBS is a parallelogram.
STATEMENTS
REASONS
1. Let 0 be the midpoint of diagonal PR. 1. Every line segment has one midpoint.
2. Under a half turn about 0, OP OR. 2. A half turn preserves angle measures and
distance.
3. PA,
RB.
3. Given.
4. PA a-' RB and in particular A -i B. 4. Reason 2.
14
5. Given.
6. Under a half turn, opposite vertexes of a
parallelogram are interchanged.
7. Statements 4 and 6.
8. A quadrilateral is a parallelogram if it is
conserved under a half turn.
5. PORS is a parallelogram
6. S-'Q.
7. AQBS-BSAQ.
8. AQBS is a parallelogram
The following theorem is proved on page 330 in Graening and Nibbelink in paragraph
form. The proof is rewritten to correct a logical error and to conform to format of a
traditional proof.
If triangle ABC is isosceles with AB
THEOREM
CM, then BL2. CM.
I
AC, and with medians BL and
C
X
The proof uses the fact that the angle bisector of the vertex angle of an isosceles triangle
is an axis of symmetry (reflection line) of the triangle.
B
STATEMENTS
REASONS
1. Triangle ABC is isosceles with
AB, AC.
2. Let AX be the bisector of vertex angle
A in triangle ABC.
3. Reflect triangle ABC in AX.
1. Given.
4. AB
AC and, in particular, B +- C.
5. BL and CM are medians.
6. AM AL.
7. AM'-AL and, in particular, M-* L.
8. BL4-*CM.
9. BL2CM.
2. Every angle has exactly one bisector.
3. The angle bisector of the vertex angle of
an isosceles triangle is.a reflection line for
the triangle.
4. Reflection in a line preserves angle
measure and distance.
Given.
5.
6. Bisectors of congruent segments form
congruent segments.
7. Reason 4.
8. Statements 4 and 7.
9. Under reflection distance is preserved.
15
The next example illustrates how transformations may be used to prove some theorems
that might be difficult to prove with traditional methods.
THEOREM Given any triangle ABC with equilateral triangles ABE and ACD as
indicated below, prove that BD
EC.
[I:
E
B
FO-
The proof will proceed by rotating the entire figure 60 degrees clockwise about point A.
It will be shown that, under this rotation, BD goes to EC. Since rotation is a congruence
transformation, BD will be congruent to EC.
STATEMENTS
1. Triangles ABE and ADC are equilateral.
REASONS
Ii. Given.
2. Angles BAE and CAD are of measure
60 degrees and AB2AE and
AD2. AC.
2. Properties of equilateral triangles.
3. If figure BEADC is rotated about point
A in clockwise direction, AB - AE and
Ad —j-AC. In particular, B —*E and
D — C.
3. The center of rotation is a fixed point
under rotation and rotation preserves
distance.
4. BD—' EC.
4. A line segment is determined by two
points.
5. 801' EC.
5. Under rotation, distance is preserved.
16
The three examples of proofs by using geometric transformation show that teachers can
still provide deductive experiences in the new senior geometry course, although the more
familiar Eucidean axioms and traditional format have been revised. The new Graening
and Nibbelink textbook also gives much attention to geometry with co-ordinates and
with vectors thus providing more opportunity for the integration of algebra and geometry.
A good example is provided by the translation. Under a translation, line segments go into
congruent and parallel line segments. Therefore every translation may be represented by a
vector, and in the plane, a vector may be represented by two co-ordinates.
It appears that the new geometry course provides an interesting challenge to teachers but
one that might very well provide more payoff than the traditional geometry course.
Mathematics teachers in British Columbia may find themselves on some unfamiliar
ground at first, but their efforts to meet the challenge could very well provide some
satisfying rewards to them and their students.
SELECTED BIBLIOGRAPHY
Frank M. Eccles. 'Transformations in High School Geometry,' The Mathematics Teacher,
February 1972: 103 and 165-169.
Howard F. Fehr. 'The Present Year-Long Course in Euclidean Geometry Must Go,' The
Mathematics Teacher, February 1972: 103 and 151-1 54.
George Gerhart. 'What Do Mathematics Teachers Think about the High School Geometry
Controversy?' The Mathematics Teacher, October 1975: 486-493.
Jay Graening and William Nibbelink. Geometry. Charles Merrill Publishing Company,
Columbus, Ohio, 1975.
Edwin E. Moise. 'The Meaning of Euclidean Geometry in School Mathematics,' The
Mathematics Teacher, October 1975: 472-477.
17
Deductiue Geometry in the
Junior-Secondary Curriculum
by ROBERT F. PEARD, Windsor Secondary School, North Vancouver
Curriculum reform in mathematics has generally made good progress over the recent years.
However, in some fields, there has been less than complete satisfaction, and with geometry, in particular, there has been a good deal of dissatisfaction. Many of us who are familiar with the problems encountered with the old Grade 10 curriculum, are pleased to see it
go without necessarily wanting the removal of geometry as a method of learning the deductive method, which is so important to all of mathematics. I, for one, believe that geometry can be used in the new curriculum to teach the deductive method and that this can
be done more easily and efficiently than in the past.
First, I should like to examine some of the deficiencies of a rigorous, strictly logical approach at the junior secondary level. In an article 'Conventional Approaches Using Synthetic Euclidean Geometry,' NCTM 36th Year Book, Charles Brumfiel cites the results of
some studies he made of student understanding ofmathematical structure and logic. His
studies, which extended from 1954 to recently, included students who had just completed
secondary school geometry and students well into their university courses. The details of
the study are well outlined in Chapter 4 of the 36th NCTM Year Book, 'Geometry in the
Curriculum,' but, I feel his conclusion is worth repeating - 'Students of 1954 who studied
an old-fashioned hodge-podge geometry had no conception of geometric structure; students of today who have studied a tight axiomatic treatment also have no conception of
geometric structure.'
Have our own students who have studied a rigid axiomatic treatment fared any better?
What understanding of structure and logic have they gained through the old curriculum,
and how might we approach the new? To answer these questions, I decided to administer
some test questions similar to those of Brumfiel's study to sorrie of our own students. Last
year, while working as supervisor on UBC's Secondary Internship program, I gave a test to
more than 100 Grade 11 students from five B.C. schools. My objectives were similar to
Brumfiel's - to determine to what extent pupils: had acquired a knowledge of the axiomatic structure of geometry and mathematics in general; learned to distinguish between
mathematical geometry and the geometry of the real world; were able to distinguish between inductive and deductive methods, and were able to distinguish between a conjecture and a proof and other ideas or concepts I consider fundamental to understanding a
deductive system or that were fundamental to the objectives of teaching a deductive system. For the purpose of my study, I chose only academic Grade 11 s who had successfully completed Grade 10 (geometry) in B.C. and who were currently taking Math 11.
The first 16 items of the test* are multiple-choice based on the above objectives. I was
aware of some possible criticisms - perhaps some of the questions were too sophisticated
or too difficult. Some items would be ambiguous at a higher level; e.g., item 10 can be
considered as theorem from Peano's axioms or as a principle of the number system; in
item 6, alternative 'A' could be correct if we applied the principle of mathematicsl induction, though alternative 'B' is the appropriate response at the secondary school level. I
* Comments refer to the test printed at the end of this article.
18
ran the results through UBC's 'LERTAP' test-analysis program. The program gives a very
comprehensive analysis of each item. In brief, it correlates each pupil's score for an item
(0 or 1 on items 1 to 16) with his total score for the test and gives a point-biserial correaltion coefficient for the item. A correlation coefficient of zero would indicate that the
item did not discriminate between high and low achievers. A coefficient of 0.4 would be
quite high and, of course, a negative coefficient would indicate that pupils who scored
well on that item did poorly on the test. For 'good' items, the correct alternative should
have a high positive coefficient, and the incorrect ones, negative coefficients. In this manner 'bad' items can be picked out and subsequently excluded from the analysis. The program also gives the mean scores of all the pupils who chose each alternative for each item
as well as the percentage choosing each alternative. Items 17 and 18 were weighted '2' for
statement and proof, '1' for statement only, '0' incorrect or no response. Hence the
achievement part of the test scored a total of 20. Items 19 and 20 were analyzed as separate subtests, scoring 3, 2, 1 and 0 for alternatives A, B, C, D respectively (alternative E
was ignored). The purpose of doing this was so that the program would correlate the
score of each subtest with the score on the main test thus giving a correlation between the
pupil's Grade 10 and Grade 11 marks and achievement on the test (of course, no one
selected alternative '1' of items 19 since they had been selected on the basis of their
success in Math 10).
Without going into too much detail, I think it is worth-while to look at some results and
their analysis (a summary of results for each test item follows the test).
The response to item 1 indicated that 57%. of the pupils were unaware of the need for assumptions in a deductive system. Yet awareness of this is absolutely essential to understanding the deductive process. Further, item 1 showed as a 'strong' item.
Item 2, not unexpectedly, was not such a strong item, but still the 33% who answered
correctly had a significantly higher mean score than the others.
Item 3 confirmed the response to item 1.
Item 4 implies that most are aware of the significance of a counter example.
I feel item 5 is particularly significant since only 17% identified this as an example of
inductive reasoning while over half falsely called it an example of the deductive method.
Further, the zero correlation implies that most of the 17% were probably guessing anyhow.
In item 9 we again see this lack of knowledge of the deductive method when 34% call a
statement that has not been proven a theorem.
Items 13-18 were similar to the items in Brumfiel's study, and the responses show the
same similarity. Even though 52% were able to identify the Isosceles Triangle Theorem
(Item 14), a full 74% were unable to state any theorem in geometry, and not one pupil
was capable of giving a satisfactory proof. As in Brumfiel's study, I received my share
of pupils who tried to prove 'through any two points you can draw a line' or an isosceles
triangle has two equal sides.
19
In the realm of number theory, our pupils fared even worse. Even though 33% identified
the statement of item 8 as 'a theorem in number theory,' 94% were unable to state any
theorem, and even the 6% who did write something acceptable confined themselves to
fairly simple theorems such as 'odd + odd gives even.'
The results of items 19 and 20 are summarized independently. The correlation coefficients
were much higher than I expected; though I'm not sure what significance to attach to this.
Further, their 'predicted' Grade 11 score correlated just as highly with the test achievement.
IMPLICATIONS OF THESE RESULTS
As far back as 1959, the Commission on Mathematics cited as one of the objectives of
math education 'a development of an understanding of the deduction method as a way of
thinking and reasonable skill in applying this to mathematical situations.' It seems to me
that we have had very little success at achieving this objective. I find this disturbing, since
I do not think the concepts involved are any more difficult than those of other parts of
math. A good deal of blame can probably be attributed to the excessive pedantry of the
rigid approach in the Moise & Downs text. This approach certainly did irreparable damage
to many students' appreciation of geometry and was correctly understood by extremely
few. Few teachers will lament its passing. However, many will wish to employ the deductive method in their junior secondary geometry courses. It is my belief that this can be
easily accomplished within the proposed curriculum, and I should like to make a few suggestions that might be helpful in achieving this objective.
THE DEDUCTIVE METHOD IN THE NEW CURRICULUM
To develop a satisfactory appreciation of the deductive method, the following objectives
should be met:
1. Develop an awareness of the need for assumptions in a deductive system.
2. Be able to distinguish between a conjecture arrived at inductively and a theorem
proved deductively.
3. Be familiar with the use of counter examples to disprove a conjecture.
4. Be able to arrive at conjectures through meaningful activities.
If a deductive approach is to be employed, we should, as much as possible, avoid proving
statements that are trivial or meaningless. It is absolutely useless to prove a statement that
is more obvious to the pupil than the methods used to prove it. For most junior secondary pupils, it is a complete waste of time to prove Statements like 'every segment has one
mid-point.' As much as possible, pupils should be able to arrive at meaningful conjectures
through their own activities. If no counterexample to a conjecture is found, we may be
able to prove it deductively, for we may accept it as true, since it sounds reasonable.
SUMMARIZING
Activity
Induction
Conjecture
Look for counterexample. If none can be found, then it may be reasonable to accept
Conjecture
Proof
Deduction-­-,
Proof is only valid in terms of definitions and assumptions.
20
The 'circle theorems' provide an excellent example of this approach. In the old curriculum, many pupils were never exposed to these, presumably because the chapter containing them was toward the end of the book, and the class had either become bogged down
in pedantry or switched to algebra before they were reached. Fortunately, the new curriculum contains the use of the properties of a circle to determine the measures of specified
segments, arcs or angles. The usefulness of these properties lies in the facts that:
1. they are easily arrived at through activity, and
2. they are not all obvious, apparent or, in general, already known by the pupil.
Once the preliminary definitions have been covered, the pupil can quickly and easily
arrive at the conjecture 'the angle at the center of the circle is twice the one at the circumference' (any counterexamples will be attributed to error or inaccuracy).
LC = 2La
Since the pupils will not, at this stage, be able to prove this conjecture deductively, they
could accept it as assumption.
The pupils are then asked to make a conjecture about the angles subtended by the same
arc.
Some, through activity, will establish their congruence as a conjecture. utners wiii eitner
consciously or otherwise employ the deductive method and reason that, since both are
twice the angle at the center, they must be congruent. The latter process is a perfectly
valid application of the deductive method in action and one that almost all pupils can
easily understand. The need for accepting the original conjecture for the proof to hold
is clear.
lfc2aand c = 2b, then a b.
If we can both prove this conjecture in terms of simpler assumptions, so much the better,
but there is nothing 'wrong' in accepting it as an assumption at this level illustrating that
every deductive system has to start somewhere with assumptions that we must accept
without proof. In general, any reasonable conjecture can be accepted and the assumption
21
used as a part of the deductive process. In this way, pupils should become aware of the
need for assumptions and be able to distinguish between inductive and deductive
methods.
The properties of quadrilaterals and the triangle congruence postulates provide another
example of this method. Once the concept of congruence is understood, the triangle congruence postulates can be arrived at inductively through activity. Again these are not
apparent or familiar to most, though the process of arrivingat them is harder than the
circle postulates. Similarly, most of the properties of quadrilaterals can be found inductively. Then, by accepting the congruence conjectures, it is relatively easy 10 prove deductively the second set of conjectures. Although this illustration would take more time than
the 'circle theorem' example, it would, nevertheless, be completed in a reasonable amount
of time. It is not necessary to spend the first three .weeks proving statements like 'every
angle is congruent to itself.' It is not even strictly necessary to differentiate between the
measure of an angle and the angle itself. These belong to a more rigorous approach that
some pupils will encounter at a later time. In the meantime, there is no need to deny the
majority the opportunity to learn what the deductive method involves.
TEST QUESTIONS GIVEN TO GRADE 11 STUDENTS
TO ASSESS THEIR KNOWLEDGE OF GRADE 10 GEOMETRY CONCEPTS
Mark your answers on the answer sheet provided.
The nature of mathematics is such that every statement in it can be proven to be true.
A. True
B. False
2.
3.
4.
Mathematical geometry is the geometry of the real world.
A. True
B. False
Every statement in geometry can be proven true.
A. True
B. False
It appears that we can express every even number greater than 2 as the sum of two
prime numbers. Some examples are:
8= 5+3
12= 7+5
20=13+7
We could disprove this statement if:
A. we could find a large number of counterexamples (examples which contradict the
statement)
B. at least one counterexample
C. two or more counterexamples
If we add two odd numbers, we always seem to get an even number, e.g., 5 + 3 = 8,
7 + 5 = 12, etc. Thus, we might say 'The sum of two odd numbers/s
always even.'
When we make a statement such as this about all the elements of a set based on true
statements about some of the elements, we are reasoning:
A. inductively
B. deductively
C. directly
D. indirectly
22
6. To prove a statement about the members of an infinite set (e.g., the statement in
Q 5), we must:
A. show it is true by applying the statement to every member of the set
B. reason deductively from generalizations
C. you can't prove a statement about the members of an infinite set
7. We can prove there are an infinite number of prime numbers by assuming there are a
finite number and showing that the assumption leads to a contradiction. This type of
reasoning is:
A. not allowed in mathematics
B. direct reasoning
C. inductive reasoning
D. indirect reasoning
8. Once proved, the statement 'There are an infinite number of prime numbers' is regarded as:
A. a definition
B. a principle of the number system
C. a theorem in number theory
9. A statement in mathematics which appears to be true but has not been proven
as in Q 4) is called:
A. a conjecture
B. a theorem
C. a definition
D. a corollary
In questions 10, 11, 12, indicate:
A. a definition in the number system.
B. a principle of the number system.
C. a conclusion or theorem which can be proven.
10. a x I = a for all real numbers, 'a.'
11. For all real numbers a, b, c,
a(b+c) = ab + ac
12. For all real numbers, X,
(X-1-½) 2 = X(X+l) + ¼
In questions 13, 14, 15, 16, indicate:
A. a definition in geometry
B. a geometric postulate
C. a theorem capable of being proven
13. A triangle with two congruent sides is called an isosceles triangle.
14. If two sides of a triangle are congruent, it has two congruent angles.
15. Through any two points there is exactly one line.
16. The sum of the measures of the three angles of a triangle is 180. Answer questions 17
and 18 in the space provided.
17. State and prove one theorem from geometry.
18. State and prove one theorem from algebra or number theory.
19. What grade did you obtain in Grade 10 Math?
E. Did not take Grade
D. Fail
C. C (including C ) C, C) B. B
A. A
10 Math
20. What grade do you expect to get or have been given for Grade 11 Math?
E. Did not take Grade
D. Fail
C. C (including C ) C, C) B. B
A. A
11 Math
23
SUMMARY OF TEST RESULTS
Questions 1-18
Option
(*correct)
%
A
B*
2
3
Co-eff.
of Con.
Mean
Score
43
57
-0.56
0.56
6.7
9.2
A
B*
66
33
-0.32
0.33
7.6
9.1
A
35
-0.35
B*
65
0.35
7.1
8.7
A
B*
C
7
82
11
-0.50
5.6
0.36
8.4
-0.15
75
A
B
C
D
17
52
17
15
0.05
0.26
-0.30
-0.18
8.4
8.7
6.9
7.4
.A
B*
C
15
62
22
-0.29
0.43
-0.35
7.1
8.8
6.9
A
B
C
D*
11
12
26
52
-0.33
-0.20
-0.19
0.40
6.6
7.3
9.0
A
14
-0.15
7.5
B
C
53
33
-0.39
0.52
7.3
9.7
Item
1
4
5
6
7
8
38
34
3
25
-0.30
-0.15
9.2
7.4
6.3
7.6
A
B*
C
27
32
41
0.00
0.47
-0.40
7.9
9.5
7.2
A
B*
C
12
53
35
-0.35
7.9
8.8
7.2
12
A
B
C
22
28
49
-0.32
-0.30
0.47
7.0
7.2
9.2
13
A*
B
C
61
23
17
0.54
-0.48
-0.26
9.1
6.4
7.1
14
A
B
C
13
35
52
-0.33
-0.32
0.47
6.7
7.3
9.1
15
A
30
42
28
-0.30
0.52
-0.30
7.2
9.4
7.2
10
11
C
24
0.41
-0.25
0.00
0.35
16
A
B
C
17
A
B
C
18
A
B
C
(weight)
0
1
2
.0
1
2
%
Co-eff.
of Con.
Mean
Score
27
22
52
-0.43
0.00
0.40
6.7
7.9
8.9
74
26
0
-0.36
0.36
0.00
7.7
9.3
-
94
4
2
-0.26
0.41
0.00
8.0
10.5
8.0
Summary of Achievement Test
Mean score
8.14/20 = 40.7%
S. D. 9.67
Normally distributed
maximum score 15/20 = 75%
minimum score 2/20 = 10%
7.5
A*
B
C
0
9
Option
(*correct)
Item
Item 19 (Grade 10 mark)
Mark
%
Co-eff.
Mean
Score
(As %)
5
S
A
B
C
F
-
13
27
59
-
0.78
6.41
-0.81
-
15/26
12.3/26
9.4/26
58
47
36
Mark
%
Co-eff.
Mean
Score
(As %)
A
B
C
F
11
19
62
8
0.75
0.42
15/26
12.5/26
10.8/26
8.75/26
58
48
42
34
Item 20
-0.50
-0.50
The Brick-Laying Problem
by PAT PRESIDENTE, Prince Rupert Secondary School
INTRODUCTION
The brick-laying problem was first presented to me in a geometry class at the University
of Victoria.
The answer is very intriguing to students and thus provides a very stimulating discussion
in either a physics or mathematics class. (I have used the problem in both situations.)
While the notation may be disturbing to some classes, it can be either simplified or discussed fully to clear up any confusion.
'I will stack bricks from Victoria to Duncan without the pile falling,' Dr. Heinrichs said,
'and furthermore, with the pile still remaining intact, I will drop a battleship on top.'
With such an opening line, you couldn't help being intrigued. In fact, the class paid 25
cents each to hear the explanation under the condition that we would get our money
back if not completely satisfied - Dr. Heinrichs kept his money.
BRICK-LAYING PROBLEM
Question:
In stacking a pile of bricks, how far can we project in a horizontal direction from the base
brick before the system will collapse?
[ -
i.e., What distance can D be before the system collapses?
Answer: D can be of infinite length.
Solution:
First we must acknowledge that this is a theoretical problem. That is, we must assume
1. That the bricks are perfectly smooth
2. That each brick is precisely the same shape and mass
3. That the center of mass of each brick is located at the exact center of each brick.
Further assumption: For simplicity's sake, let us assume that each brick is of unit length
and of unit mass.
Definition: Center of mass (c o m)
The center of mass of a set of K objects, X is given by:
X=X1m1+X2m2+....+X1m+....+Xm
m1+m2+....+m.+....+mk
25
Note: the center of mass is expressed in terms of the distance of the center of mass from
the origin of a given co-ordinate system.
I.e., X is the distance from the c o m of the th object to the origin of the co-ordinate
system.
m is the mass of the 1 th object.
To define the X distances, we will draw a linet to be our origin. (See diagram.)
£
Definitions:
ai We will define a i as the distance from.L
to the c o m of the system containing i
bricks.
I.e., it shows us where to place the + 1
brick to keep the system stable.
/
i
3
5-
d 1 We will define d as the additional distance from ,€. that the lower brick is
from the preceding one.
I.e., d 1 = a —a i_i with a0 = 0
Note: the stacking is done in reverse (i.e., from the top to the bottom).
___
*
^
x1^L..,i d,
":za
=;
•j/•/ =:
/
. • .x
d:4.
.t.
L3
3
3
3
1/6 is an important result since if d 3 were 1/8, d 4 were 1/16, etc. —the series would have
converged, i.e., i".
I will now give a general expression for calculating a:
-,
Now we can calculate a 4 using this formula rather than the old procedure, (shown at left).
=+(&)= 4/.
24L
26
Note: with the placing of the 5th brick, we are already past D
1 brick length.
=
brick lengths.
I.e.,
d1.32_^TL
,
eta.
D=d,d+d3-...
NOAJ
/
/
L
/
I
-'--,j
-'-••••
3; =
+
-
e-te4
But this is the harmonic series which can be shown to diverge, i.e., it becomes infinitely
large and does not approach a limit.
This can be illustrated by showing that a series that is smaller than the harmonic series
diverges.
To obtain this series, replace 1/3 with 1/4, replace the 4 terms 1/5 to 1/8 with 1/8,
replace the 8 terms 1/9 to 1/16 with 1/16, the next 16 terms with 1/32, etc. (each group
adds to 1/2).
I.e.
I
1.
I
/
/
7 51
/
4-
++
V2.
T'
/0
+
Since this series becomes infinitely large, so must the larger harmonic series. Hence D
becomes infinitely large.
27
PRIMARY MATHEMATICS TEACHING
The Hundred Square
by OLIVE M. STEWART, Supervisor of Primary Instruction, Coquitlam
DEVELOPING AN UNDERSTAND INGOF BASIC RELATIONSHIPS
THROUGH THE USE OF THE HUNDRED SQUARE
Preparatory Work - Kindergarten or first year
A. Suggested activities for developing an understanding of:
left-to-right progression
- before and after
1. Use a stepper mat made on a plastic strip divided into squares. Have a pile of varied
shapes of different colors. Ask a child to put a red circle in the first box, a red square in
the next box, a red triangle in the next box, etc. (attribute blocks are excellent for this
activity).
Another activity with a stepper mat involves before and after. Teacher puts a shape in one
box and asks a child to put a shape in the box before it and another child to put a shape
in the box after it.
If youwish, each child can have his own cardboard strip and small shapes.
L I
I
I
I
III
Teachers who use small individual math boxes may wish to include the strip and shapes
in the material used in the boxes.
A variation of the strip activity can be developed by sticking colored gummed circles in
the squares. Ask children to touch the circle before the red one, after the red one, etc.
Another variation is to have children fold a paper into four parts to make apartment
buildings.
EB ED ED EB
In each part, the children make a window in the color you choose. Have children put
their finger in the apartment building with the red window. 'What comes before? What
comes after?' At a later stage: 'What comes first? second? third? fourth?'
28
2. A simple game for developing left to right can be made from a long squared strip.
Two children play the game. The die is a cube with the four shapes, one on each side, and
two sides left blank. The players move little toy cars or men or plastic discs. Child tosses
the die and moves car to appropriate shape. If the die turns up blank, car stays still. Aim
is to see who reaches the end first.
3. Use pegboards. Have children fill in a row from left to right. 'First use a red peg, then
-.
a yellow peg, etc. What comes before the yellow peg, after the yellow peg?'
4. Have children draw a road from left to right. 'Make a tree above the road. Move along
the road. Make a house. Move along the road. Make a man. What comes before the house?
What comes after the house?'
5. Use a clothesline. Hang up a sock, a shirt and a dress. 'What comes before the shirt;
after the shirt?'
Use later with numbered socks What comes before the 2? - after the 2 7
199
6. Make a large staircase in squares, with a space between each step. An old plastic
tablecloth is ideal.
Let children walk on steps and call them out. Have children walk up the steps and tell
you which is bigger, smaller, etc. Have children tell you what comes before 3, after 3, etc.
A related seatwork activity can be made from a staircase on tagboard.
I.
MOFO1.
Jetc
Children place colored counting discs in the circles.
B. Using the strip for numbers to 10:
1. Children fill in numbers 1 to 10 on cardboard strip. 'Put your finger on 2. What
comes before? What comes after 2? Repeat up and down the line.'
'Close your eyes. What comes after 2? Open your eyes and see.' Repeat many times.
'Put your finger on 2. jump over 3. Where are you? Let's play leap-frog and say the
numbers as we go: 2, 4, 6, 8, 10. Let's play leap-frog backward: 10, 8, 6, 4, 2.'
29
2. Teach +1 going up the line to 10. At first, let children use their finger on the line.
Then call out numbers and have children think what comes after. Teach +2 similarly playing leap-frog.
-
3. Teach —1 going backward. Teach —2 going backward. For understandingp/us and
the following rhyme may help.;
minus
-
When I see Plus I move ahead.;
With minus I go back instead.
- Children will benefit from using a staircase made on cardboard (as in A. 6) to help them
understand the number values as well as position.- - Have the children use their fingers to
move up and down. Staircases on cards can be used very effectively with 'a small group
of children working around a table - each child using finger on own card.
4. For facts in the number families, provide squared paper and have children color the
various combinations. For each fact have the four stories written.
E.g. _______
3+1=4
iciitri.
'1+3=4
4-3=1
4-1=3
(Note: in the following activities, the term lOOsquare refers to a 10 by 10 array of
squares. See example on the 'Tim Turtle' drawing.)
Can you find a pattern
in Tim Turtle's
100 square?
.1
Tell about your pattern:
30.
C. Using the squares for counting in various ways to 100 and beyond.
1. Give children many opportunities to fill in numbers to 100 and look for interesting
patterns, e.g., counting by 5s and seeing the 5 or 0 endings.
2. Squares can be numbered in various ways, e.g., starting at 100 and numbering down.
Have children leave some squares blank, pass the sheet to a friend and have it filled in.
3. Use the 100 square to fill in numbers from 100 to 200, etc. Circle to show counting
by 2s, 5s, etc., from 100 to 200.
D. Using the squares for addition and subtraction through the decades.
1. Provide each child with a curtain ring, transparent plastic disc, cardboard ring or
commercial plastic square (available from Moyer).
2 + 1 Children circle the answer.
Call out:
12 + I Children circle the answer.
22 + 1 Children circle the answer.
Proceed through the decades. Use similarly for subtraction.
2. For facts involving bridging (moving into next decade), children will need much practice using the ring in oral, work. Once children have the idea, give them much seatwork
6 + 7 = 13
practice. E.g.,
16+ 7 =
26 + 7 =
Have children continue on their own.
E. Using the squares for factoring, multiplication and division.
1. Children can fill in the squares in the study of a number to show different arrays:
E,6
1x66
6x16
2x36
3x26
66=1
6+1=6
6+3=2
6+2=3
2. Counting and repeated addition can be related to multiplication:
123J
q-56
7 9
10 1 11 1 12 1
1x33
2x3=6
3x39
4x312
L tc.
3. Multiplication/addition principle. E.g., 4 x 8 = (4 x 5) + (4 x 3)
Have the child cut here to see that
4x8(4x5)+(4x3)
1
31
4. Multiplication facts can be stressed through use of a window card, e.g., for 7s, the
child can circle 7, 14 0 21, etc., and then make a duplicate to cover the first sheet. On the
covering sheet, have the child cut out 7,-14, 21, 28, etc., and place over first sheet. Child
sees an interesting pattern.
F. Using the squares for perimeter and area in metric measures.
Make the squared paper using centimetre lengths and have children show you a rectangle
five centimetres long and four centimetres wide. 'How far is it around the rectangle?'
'Make as many shapes as you can that have 10 square centimetres in them.'
G. Using the squares for review purposes children will enjoy making addition (and
subtraction), multiplication (and division) tables using the squared paper. E.g.,
X
.z 2 3 i-
JOOOQ
[jjOi23
Lf.
[2IIO26
Stress commutative property of addition and multiplication: e.g., 3 x 2 2 x 3.
H. Using the squares for discovering 'square multiplication facts.' Have children use the
squared paper to discover the 'square facts' - 1 x 1, 2 x 2, 3 x 3,4 x 4, etc.
I. Using the squares for developing an understanding of prime and composite, odd and
even numbers.
Let children try to show the numbers beyond 2 as rectangles. Prime numbers have only
two factors - 1 and the number itself. Rectangles are composite numbers (composed of
factors).
J. Teachers may also wish to use the squared paper for bar graphs, co-ordinates,
fractions, etc.
TEXTBOOK ORDER FORM
There have been reassurances from the Department
of Education that the June order forms will be
corrected in respect to Grade 9 and 10 textbook
ordering. The section on '25% of total enrollments,'
will be changed to '25% of total enrollment for
each title.'
32
The Child's World: Mathematics
by WERNER LIEDTKE, Faculty of Education, University of Victoria
[Editor's Note: The ideas presented in this article are an elaborabion of those included
as an appendix to Dame!! Rhea's article, Vector, Volume 17, No. 2, p. 12-13.1
Just as some cute sayings that young children utter may show the existence of ideas quite
foreign to the adult's mind, the solutions they supply to a particular problem can provide
a lot of insight into their 'level' of thinking. When faced with a problem, the majority of
young children are very willing and eager to attempt to solve it. They show endless enthusiasm and almost never fail to supply a solution. Many of the solutions they provide indicate or support the idea that young children are not miniature adults and that their view
of the world is unique.
In the paragraphs that follow a few tasks are described that can be used to show what the
young child's 'world of mathematics' is like. These ingenious tasks, devised by Piaget, can
be used to gain some insight into how children think. Posing the question, 'How do you
know?', noting responses to this question and observing the child's behavior can be a
learning experience for both the adult and the child. The child will find new ways of
looking at 'familiar objects and the adult can discover a more effective or meaningful way
of communicating with the young child.
Quantity: Two identical glasses containing lemonade, juice or a liquid of some sort are
presented. One of the glasses contains a bit more liquid than the other. When asked to
choose a glass, most young children will take the one that has more in it. However, children will change their minds once the liquid from the glass that contains less is transferred into a taller, narrower glass.' The simple act of pouring the liquid into a different
glass will, according to them, 'make it more.'
Typically, the problem described above is begun by presenting two identical glasses and
requesting that they be filled with the 'same amount' of liquid. In this setting, a few
children will spend a great amount of time attempting to satisfy the given condition.
Repeatedly a few drops are added and subtracted here and there, and the procedure can
turn out to be a real test of patience for the observer.
Children will reach the stage when they think that the act of pouring the liquid into a
taller and narrower glass will not make it 'more.' The response to the question 'Does this
one have more, that one have more, or is there the same amount to drink in both of
these?' often consists of an expression on their faces that seem to say, 'That's a silly question to ask.' One observant six-year-old emphatically claimed that the taller, narrower.
glass contained less. When asked, 'How do you know?', she pointed to the table and
stated, You spilled a few drops.'
33
Number: Most young children know how to count. Counting, for some, seems to be a
status symbol as indicated by the comment, 'I can count further than you can.' However,
for many preschoolers, counting is a rote or copied rather than a rational procedure.
While counting a number of objects, children fail to match number names with objects
in a one-to-one fashion. As a result, six buttons may be labeled as five or seven depending
on whether a button has been skipped or a space between two buttons has been included
during the counting procedure.
Six red counters were placed on a table. One boy was asked to find as many blue counters.
The problem was solved correctly, and he was asked, 'How do you know you have as
many blue counters as red ones?' His response consisted of counting all of them. He ended
up by saying, 'thirteen' and then announced, 'There;'they are the same.'
Most young children will agree that two identical containers hold the same number of
beads after they are dropped simultaneously, one by one, into the containers.
Once the beads from one container are transferred into a different container, there will
be more or fewer beads, depending on whether the new container is taller and narrower
or wider and shorter. One young girl insisted that the taller, narrower container held more
beads, and her justification consisted of the simple statement, 'I can see.' According to
her, if two necklaces were to be made from these beads, the one made from the beads
in the taller container would be longer. Two pieces of string were given to her, and she
proceeded to make two necklaces. These looked the same length, and she seemed to
realize this. However, when she held them up to show the result, she would hold them in
a position to make the one she wanted to be longer look longer.
For most young chi!dren, the simple transformation of spreading out or bunching up,
performed on one of two groups of equal objects will make the objects become more or
fewer respectively. Some will go as far as to say that 'this four is more' when talking
about the circles in the following arrangement:
0 DOD'
Length: When asked to somehow show that two sticks or strips of paper are the same
length, young children usually fit the two objects together and then compare the endpoints. When a transformation is performed on one of the objects, such as moving it to
a differentposition (- - . ; or ....L. ), the sticks are of different lengths.
An interesting conflict can be created by asking a young child who thinks that moving
one strip of paper makes it longer to take a pair of scissors and to make the strips the
same length. The suggestion of making a cut is all too eagerly followed. However, as soon
as a cut is made, many children will notice that something has gone wrong. 'Oh, oh cut it in the wrong place,' is a typical comment. Some children may attempt, in an inconspicuous way, to put the cut pieces together again. Occasionally a child becomes very
frustrated, realizing that something is wrong but not knowing what it is; and it seems best
to change the topic or task quickly.
Length is looked at in terms of endpoints and not as a characteristic displayed by an
object. The following example illustrates this point rather well. A stick and apiece of
34
).
string are judged to be of the same length when the endpoints are coterminous ),
it
is
judged
to
be
longer.
However,
when
the
When the string is extended ( ), it is the same length as the stick string is returned to the original position ( again.
Measurement: To gain some insight into how well young children understand measurement, the following problem is presented: Blocks are used to build a tower, and the children are asked to use different blocks, to avoid matching, and to build a tower just as high
as the given tower, some distance away from it. After the tower is completed, the children
are asked to show, in some way, that the towers are the same height. Sticks and pieces of
string are made available. )^l
To prove equality in height, most young children will simply rely on an 'eyeball' test.
Some check by lowering their heads and partially closing one eye. Others will place one
hand at the top of one tower and then carefully move the hand, parallel, to the table top
or floor, across to the other tower. The children see no need to use the sticks or pieces of
string. One boy followed the procedure of moving his hand across after every block that
was placed into position. Sometimes the procedure was repeated two or three times for
the same block. He appeared to attempt to average out the many dips that had occurred
each time his hand was moved across.
When the children are asked to use the sticks to show that the towers are the same height,
they will use the sticks in a horizontal fashion, from tower top to tower top. For these
children, duplicating on the floor the height of a tower that is standing on a table, and
hen showing that the two towers are the same height presents a rather difficult problem.
One girl solved the problem by taking a stick, placing it horizontally at the top of the
tower on the table, and slowly moving it toward the edge of the table. There the stick was
changed to a vertical position. Movement was toward the floor. When the approximate
height of the lower tower was reached, the stick was changed back to a horizontal position. It was kept like this until the tower was reached.
When attempting to show that two towers are the same height, some young children are
able to consider both, the top and the bottom of the towers. These children will place
one hand at the top, the other at the bottom, remove the hands and carefully transfer the
simulated height. A great amount of confidence was displayed by the 'engineer of the
future' who placed one hand at the top and the other at the bottom of one tower. Carefully he removed both hands and inspected the distance between them. Then he dropped
his hands. Having walked over to the other tower, he again simulated the height of the
tower he had measured., held his hands against the other tower and proudly announced,
'Yes, they are the same.'
Horizontal: While sitting around the supper table many young children will carry on a
converstaion, casually holding a full glass of juice or milk in their hands. When attempting to take a drink from a full glass, frequently they will tilt the glass too much. In both
cases, the results are disastrous. Even after numerous experiences with liquids, on the
beach or in the bathtub, young children have no idea how liquids behave.
V
A partially filled bottle is presented. On a drawing of the bottle, the child is then asked
35
to draw a line to show where the water comes and to put an X in the water. To gain some
insight into how the child thinks the water behaves, the bottle is hidden in a bag and various positions are shown to the child. For each position a drawing is presented and the request for a line and an X is made each time. Typical responses include:
X
X
No wonder there is a no need to worry about spills. If that is how liquids behave, a glass
will have to be tilted rather vigorously to get the juice or milk out of it.
Vertical: A lack of understanding the concept of vertical becomes apparent when young
children are asked to draw a chimney on the roof of a house, or a tree on the
side of a hill. The results are predictable. However, there are quite a few young
children who seem to know how to draw a tree on a hill correctly. Perhaps
they were told how by an adult or by a older brother or sister, since their
responses to presenting other details, such as a fence, a power line or a•
doghouse, are often done incorrectly.
1^r
Time: When a group of adults are asked to close their eyes and to indicate when they
think one minute is over, the responses usually range from fifteen seconds to two minutes.
One can easily imagine the difficulty a young child may have in interpreting such state
men s as 'We (you) have one more minute to ... ,' 'We'll go in one minute,' or 'We'll go in
five minutes.' After being told to 'wait one minute,' one young boy responded by saying,
'No, your minute is too long.'
t
The task consists of having children slowly transfer objects from one container to another
while watching a sandglass empty or a second hand go around once. The problem is repeated, but the objects are to be transferred quickly. Then, the children are asked to compare the movement of the sand or the second hand for the two tasks. Most young children will claim that the timers advanced more rapidly for the second task. This is an interesting interpretation of time, but perhaps not too different from adults' statements
'time flies.'
The responses and justifications young children present illustrate, time and time again,
that they do rely heavily on perception. Children can consider only one variable at a
time, and their thinking is irreversible. There are other problems that could be used to
illustrate these points. Tasks that deal with such concepts or ideas as distance, area,
volume, class inclusion or the quantifiers: all - some none - one - can be used to
show that young children are not like adults in their thinking.
References:
Copeland, R. How Children Learn Mathematics.
(2nd Edition) Toronto: Collier-MacMillan Canada
1974.
Elkind, D. 'Misunderstandings about How Children Learn,' Prime Areas, Vancouver: BCTF, Spring
75.
Lavatelli, C. Piaget's Theory Applied to an Early Childhood Curriculum.
Boston: Center for Media
Development, Inc. 1973.
Nelson, D. and Liedtke, W. Mathematical Experiences in Early Childhood.
Toronto: Encyclopaedia
Britannica Publications Ltd., 1972.
36
-
SAMPLE MATHEMATICS ACTIVITY CARDS
FOR USE WITH HUNDRED SQUARE
These can be pasted on attractive shapes, answers put on back and then laminated. You
may wish to put answers on separate cards and put these in a file box for children to use.
Fill in the number squares from 1 to
2
100. What numbers are 2 more than:
1
Fill in the number squares from 1 to 100.
7. 98
4. 28
What number is missing in each row below? 1. 4
8..49
5. 35
2. 18
Draw rings to
0
9
8
1. 7
9.60
6.15
3.10
show you can
o
6
4
2. 2
Can you do these?
count by 3s
0
60
40
3.30
from 3to36. 4+2=0 18+2=0 29+2=0
0
53
43
4.33
Make some +2 stories.
6
0
8
9
5. 10
Fill in the 100 squares from 1 to 100
Fill in the 100 squares from 1 to 100 4
3
Use your squares to help you draw the
Can you see a pattern here?
arrows:
3
1
1
10
3
7
80
Finish the pattern. Use your squares.
Fill in the squares from 1 t 100. Use
5
an egg-timer and see how many numbers
you can write following this pattern:
4,6,8,10 1
,
Try these with the egg-timer:
1.Upini0s from 2.
2. Down in 1 O from 75.
3. Down in 3s from 36.
4. Up in 4s from 4.
5. Down in Ss from 100.
1
91
9L1.
Fill in the squares from 1- to 100.
6
Put a ring around the answer to:
5-3=o
15-3=0
25 - .3 = o.
35-3=0
Fill in the squares from 1 to 100.
7
Can you find the answer to:
Can you finish the stories up to 95 - 3 =
0 ? Write them on the back of your
paper.
Fill in the squares from 1 to 100.
8
Can you find the 'answer to:
6+5
16+5
26+5
36+5
12-3=0
22-3=0
32-3=0
42-3=0
=0
=0
=0
=0
Can you finish the stories up to 86 + 5 =
El ? Write them on the back of your
paper.
Can you finish the stories up to 92- 3 =
0 ? Write them on the back of your
paper.
37
TEACHING GRADE 9 AND 10 MATH
Using'Mathein -a tics for a Modern World
by WILF BAXTER
[Editor's Note. Wi/f Baxter is a co-author of Mathematics for a Modern World, one of the
recommended texts for the revised Grade 9-10 math courses. The following is an article
based on the seminar he gave at the 1975 Summer Cánference. ]
A. Some Basic Ideas
1. Students should be given ample opportunity to work in problems during class. These
include developmental questions in the presentation of new material as well as the regular
assignments.
2. The teacher should employ a wide variety of teaching techniques. They need to be
carefully selected, and they depend on the topic involved as well as the nature of the
class being taught. The methods could range all the way from independent study to
Socratic instruction and brief lectures.
3. Teaching mathematics should be done as much as possible with understanding, but
on occasion, it may be necessary to omit a proof. However, the rigor of a deductive
system should be avoided. except for the capable student, and the emphasis placed on
applications.
4. Students should be provided with limited, short-term goals, and the presentation of
material spaced to ensure success. Usually, the difference between the average and gifted
student is found in the time required'to assimulate new concepts.
5. Teachers have an important role to play in integrating a course of study. On a daily
basis, this means placing each section in the context of the overall unit, and in the longer
term, it means providing a 'birds-eye view' of the course at strategic intervals and especially at the conclusion.
6. Applications of mathematics to practical situations should be stressed as much as
possible. These can be used for motivational interest in the development of a topic as
well as in the assignment.
7. Teachers are accountable to develop skill levels in their students. There is littledoubt
that drill is necessary to give students enough confidence with an idea before it can
become a building block to further concepts.
8. The enthusiasm of the teacher who is interested both in the students and the field of
teaching is the key to great teaching. In other words, be a practitioner not merely an importer of skills.
B. The Development of Mathematics for a Modern World, Books 1 and 2.
These textbooks were created to answer a need in a teaching situation in Grade 9 and 10
mathematics. In the mid-sixties, texts that provided insufficient exercise material were
adopted for use in our schools. For a year, we produced supplementary sheets and then
decided to reproduce our own material and replace the complete text. Keeping a couple
of weeks ahead of the students, we managed to beat them to the end of the next year.
Feedback from the harried teachers dictated considerable revision, and over a period of
five years, we developed the books that were originally published in 1970 and 1971.
38
Some fundamental principles were established in this process, which reflect our philosophy of teaching as outlined earlier.
1. The textbooks were to have an abundance of questions, because, if it is nothing else,
a textbook is a source of questions.
2. Provision for a wide variety of teaching styles was to be made.
3. The writing was directed to the student personally and was not to be only a mathematical explanation of a topic.
4. Every effort was to be made to enable students to develop and maintain basic mathematical skills.
5. Each section was to concentrate on one topic only and, to encourage achievement of
limited goals, was to take no more than one or two class periods to complete. Each unit
was to be the work of one week.
6. As many practical applications as possible were to be included.
During the past 1 5 months, the original books have been completely revised. This was in
response to (a) suggestions of the teachers who have used the texts, (b) the demands of
converting to a book that would meet the metric standards laid down by the Federal
Government, and (c) recommendations from the curriculum committee.
C. Teaching New Material from the Texts
Most sections are written so that any one of the three basic techniques can be used.
1. Self-learning
2. Teacher-guided learning
3. Traditional Socratic instruction.
Refer to Unit 2 on solving equations in Mathematics for a Modern World, Book 1. Following a section on solving by inspection, the skills of solution by adding and subtracting
are covered in Section 2.2.
Lesson Plan A
1. Review solution by inspection through taking up the assignment from the previous
day.
2. Introduce the new topic by posing a more difficult example.
3. Assign class the task of working through A on pages 20-21.
4. Check that the basic idea is clearly understood.
S. Assign B and check.
6. Oral discussion exercise page 22.
7. Assignment from questions 4-8, page 23.
Lesson Plan B
1. Asin plan A.
2. Work through A and B orally with the class with two or three pauses for the class to
work on examples at the seats where appropriate.
3. Oral Exercise page 22.
4. Assignment from questions 4-8, page 23.
Lesson Plan C
1. Asin plan A.
2. Teach a Socratic lesson with textbooks closed using example of your own choosing.
3. As in plan A.
4. As in plan A.
39
Consider another section, this time taken from Unit 18 of Mathematics for a Modern
World, Book 2. This is the introduction to trigonometry, and Section 1 introduces the
sine, cosine and tangent of actue angles using 900 triangles. In Section 2, the ratios of
special angles 30 0 , 45 and 60 are discussed, and this leads on to the use of tables.
Section 3 covers applications to finding unknown length, distances and angles.
Teaching Section 2 would probably require two 35- to 40-minute periods, but only one
of the 70. to 80-minute length. This lesson provides the teacher with an excellent opportunity to integrate ideas from algebra - work with radicals, geometry - the isosceles and
equilateral triangles, and trigonometry. Here again the teaching method can vary from independent study through to teacher-centered instruction.
You will note that the general definitions of the trig ratios follow the actue angle definitions in this text. This order, which is a change from the first edition, results from the
advice of teachers using the book. The general definition may go well coming first for the
mathematically gifted, but this has not proved to be so for the average student. Another
advantage is the speed with which you can reach to practical applications.
D. Drill and Review
This heading probably looks quite old-fashioned, and you would be quite correct if you
inferred a shock tactic on my part. We are talking about core mathematics - the acquisition of the fundamental skills that enable students to move with confidence into the
study of higher mathematics. Does this require drill? Certainly. Does the teacher have a
role to play? Definitely.
It seems to me that some of the primary responsibilities of a teacher have been ignored in
the excitement of getting into student-centered learning. We are accountable -there's a
fashionable term today that was well understood yesterday - and our task is to know the
levels our students reach. More than that, our task is to do something when students fail
to achieve their potential.
To be specific, in the texts, we have included three assists to review,
1. a regular series of maintaining basic skills exercises (MBS)
2. Summaries at the end of each third of the books,
3. Review exercises.
The intention of the MBS exercises is to review with the student the skills that should
stick and not have to be retaught. Any teacher worth his salt will want to expand on
these. Although carefully constructed, exercises should serve as just an example of the
type of material every teacher should prepare to check that learning is taking place.
The summaries are also only an example for teachers to follow. I feel that' a primary task
of a teacher is to impart the skills of study and organization of course material to students. By graduation from secondary school, each student should know how to summarize a course.
The review exercises provide an opportunity for a wide variety of applications within a
single series of problems. There are three such exercises in each book, with approxicontinued on page 49
Ell
mately 40 questions in each. There seems to be little doubt that mastery in mathematics
is best illustrated by the solution of problems as opposed to a lengthy verbal discussion of
them.
E. CONCLUSION
We have attempted in these texts to produce material that the average student can understand. The format, established with that in mind, results from our basic beliefs regarding
how students learn mathematics. We trust that you will find in them the flexibility
necessary to appeal both to the individual differences of your students and to your own
teaching strengths.
1 ereAnnonce
15e Rencontre internationale du
Centre Beige de Pêdagogie de la Mathématique
sera consacrée
au theme
SITUATIONS J EDAGOGIQUES
dans les
locaux
du DOMAINE DE HAUT-ENHAIVE
et de I'ATHENEE ROYAL DE JAMBES
JAMBES
les
24-27a6ut 1976
Le congrs sera precid d'une rencontre de Mthodo/ogie de ía mathematique (groupe de contact du FNRS) le lundi 23 a6ut 1976 au Domaine de
HautEnhaive.
For further information, contact: CBPM,
Avenue Albert 224,
B-1180 Brussels, Belgium.
49
INTERMEDIATE MATH TEACHING
A Unit on Numeration Systems
by PETER MAKE IV, A.I. Collison Elementary School, Nelson, B.C.
[Editor's Note: Peter Makeiv presented a workshop on 'Curriculum Development in
Intermediate Grade Mathematics' at the 1975 Summer Conference. The following material on number systems was used as an example of a well-planned unit in mathematics
(wit/i some excellent overlap into social studies).]
Introduction: Too often, studies of numeration and base systems employ trivial examples
of positional value and involve Hindu-Arabic numerals. Such is not the case in this presentation. Furthermore, the numeration systems presented are real.
I. THE ZAMBIAN NUMERATION SYSTEM.
A. Zambian Numerals
The Zambian numeration system currently in use is an example of perhaps the very
simplest type of system one can conceive symbolically. Only one numeral is used. It is a
simple verticial stroke usually made with a finger or a stick on the earth. These 'tallies'
are grouped by tens so that counting can be easier. There is no concept of place value and
no zero symbol.
Some Zambian numerals follow:
II
111111111.1 Il
lIllIllI
)IllI 11111
II
2.2-
Note that basically tally marks are used and that the tallies appear in groups of 10.
Suggested Pupil Activities
1. Weather permitting, take the pupils outside without pencil or paper. Have them collect a pile of stones or pebbles and direct them to draw the numerals of their collection
in the dirt or sand. Encourage children to use their fingers, sticks, or stones to make the
markings. (This should provide pupils with the 'feel' for this sort of symbolization. Tallies
should be grouped into tens.)
2. Sample seatwork exercises:
a. Express the following with Zambian numerals:
1) 15
2) 27
3) 75
b. Express the following using Hindu-Arabic numerals:
1) Ilaii:,j,i
2)/il
3) IIliiii,ii
50
I)1IlIII?I
i i,i,in,,
/1/1/1/
c. Name one numeral in the Hindu-Arabic numeration system that is the same in the
Zambian system.
d. Would it be easier to write 'ninety-six' in the Zambian numeration system or in the
Hindu-Arabic system? Why?
e. How would you add (subract, multiply, divide) using Zambian numerals?
B. Zambian Words for Numbers
Despite the simplicity of the numeration system the Zambian displays on paper or on the
dirt, his names for numbers are perhaps more systematized than ours:
9 - fisano nafine
1 - chimo
10— ilumi
2 —fibili
11 - ikumi nacimo
3 - fitatu
12 - ikumi nafibili
4 -fine
20 —amakumi yabili
5 —fisano
100 - unwanda
6 - fisano na cimo
1 ,000 - umukama
7 - fisano na fibili
8 - fisana na fitatu
One thousand is the highest possible number; ie., the Zambian feels that there is nothing
greater than one thousand objects. Infinity is inconceivable.
Suggested Pupil Activities
1. Lead pupils to discover the naming system, with such questions as:
a. With what word do the numbers six, seven, eight, and nine begin? How does this
compare with five?
b. What words in the numbers six, seven, eight, and nine compare with the first four
numbers?
c. Fi is a prefix which means 'more than one.' What do the prefixesna andya mean?
(Pupils should realize that 65+1,75+2,1110+1,1210+2, and soon.)
2. How do you think the following numbers would be named?
c. 120
a. 25
d. 36
b. 101
3. How do the names for the numbers from 20 to 29 compare in the Zambian numeration system with the Hindu-Arabic numeration system? How do they differ from 10 to
19?
C. In Summary
The Zambian numeration system contains a very systematic naming system for numbers
by fives and by multiples and powers of ten. It lacks place value and has only one symbol,
therefore, no zero: This system basically employs the additive principle using tally marks.
II. THE ANCIENT EGYPTIAN NUMERATION SYSTEM
The Ancient Egyptian numeration system differs from the Zambian system basically only
in its notation. It, too, lacks the property of place value and does not have any need for
a symbol representing 'nothing.' It does, however, have an added feature in that its symbols range in multiples of ten.
51
Egyptian Numerals
Hindu-Arabic Numerals
(staff)
(yoke)
(scroll) or Q (coil)
g or
(lotus flower)
(crooked
line or finger)
(
(fish)
91 (man in astonishment?)
1
10
100
1)000
10,000
100,000
i,000,000
(1
9
In writing, no more than four similar symbols appear together in one line, and with an
odd number of symbols, the larger frequency appears on top, i.e.,
"
'II
ii
:111
II
i/I
/1/i
iii
i/I
2
3
4
5
6
7
8
9
It is also noticeable that some sense of equal grouping was employed as in the number
instead of
or fill
fl
nine; i.e.,
The Egyptians used an addition principle in their system of writing numerals; the order did
not matter, as 12 could be represented as either ii () or fill
Ninety-nine was represented as:
/1 /7/1//I
(?afj ,; I
t1t11Hii
or as
/// /')/I/)
i,''
/11/3/7/)
Suggested Pupil Activities
1. Give pupils exercises in changing numerals from the Hindu-Arabic system to the
Egyptian numeration system and vice versa, for example:
a. What do the following numerals mean?
1) ii',
5) G)CE)
2)1?!
6)5c),),#7II
0
3)flf)fJi,,
4) 91
8)9999/1,,
b. Express 345 with Egyptian symbols.
9 9 9 jgg
c. Solve:
nriii
2. Sample seatwork exercises:
a. How many unique symbols are there in the first ten numbers? How many are there in
the Hindu-Arabic system?
b. How many more unique symbols are introduced from 11 to 1,000,001? How many
more unique symbols are introduced in the Hindu-Arabic system from 11 to 1,000,001?
c. Write any two symbols in the Egyptian numeration system (() I , for example). Write
them in a reversed order. Does this change the value of the number? Do the same with
any two Hindu-Arabic numerals. Does this change the value of the number? Why?
d. If the Egyptians had a symbol for 4 zero,' where would it be used? Why then do we
have a zero?
52
In Summary
The ancient Egyptian numeration system employed the additive principle and contained
several symbols, each for a differentpower of ten. Thus the number of symbols increased
the closer the number came to the next power of ten, therefore making the system relatively awkward to write. It lacked place value and did not require a zero.
Ill. THE ROMAN NUMERATION SYSTEM
Like the Egyptian numeration system the ancient Roman system lacked both place value
and a symbol representing zero. It too, had a limited number of symbols based primarily
on two numbers, 1 and 5, increased by powers of ten; i.e.,
1 = 1; V 5; ' X 10; L = 50; C = 100; D = 500; M= 1000.
The Roman system was unlike the Egypitan system, however, in that the order of placement of numerals was important, for both the addition and the subtraction principles
were utilized in writing the figures. IV meant 5-1, while VI meant 5+1. Further 'subtraction' numerals follow:
ix =9
IV =4
XC=90
XL 40
CM=900
CD 400
Upon closer observation, one notices that only either the same power of ten or the closest
power of ten is subtracted, i.e.,
400=5(102)_102
900=10 3 _10 2
90=10 2 _ 10 1
40=5(10')-10'
4=5(10') —1(10°)
9=10 1 -10 0
Unlike the Egyptian system, which repeats the same numeral four times, the Roman system repeats the same numeral up to a frequency of three times.
Roman
Egyptian
30=XXX
3O = '
40=XL
40 =
/) t-1 t_1
0—L
50 ' f' '1
A line placed over a Roman numeral increases its value 1,000 times.
X = 10 but X = 10,000
E.g.,
CXI 91 and.i 91,000
Suggested Pupil Activities
1. Give pupils exercises in changing numerals from the Hindu-Arabic system to the
Roman numeration system and vice versa:
a. What do the following numerals mean?
7) MXII
4) LXVI
1)11
8) MMCCXI
5) CCLXXVIII
2) VII
9) CCLXXXIII
6) DXII
3) XXVII
b. Express 987 in Roman numerals.
2. Sample seatwork exercises:
a. Write all the numerals in the Hindu-Arabic numeration system. How many are
there? Then write all the numerals in the Roman numeration system. How many are
there?
b. Write ninety-nine in the Egyptian system, the Roman system, and the Hindu-Arabic
system. Which takes up the most space? Which, the least?
53
c. Write one thousnad in all three systems. Which is the longest system numeral?
d. Most textbooks use the Roman numeral system in certain places. See if you can
locate them.
e. Write the numeral 19 in both the Egyptian system and the Roman system. Now write
an equation for arriving at the number for the Egyptian system in terms of tens and ones.
Do the same for the Roman system. Are the operations shown in both equations the
same?
f. When we write 23, we mean 20 + 3. In what respect does this resemble the plan used
by the Egyptians and the Romans?
g. Solve:
1)
XXII
2) xx(/
tXf
___
IV THE BABYLONIAN NUMERATION SYSTEM
This is an excellent example of an ancient system that utilized place value, but had no
zero symbol. Place values alternated. The second place is ten times the first, the third
place is six times the second, the fourth place is ten times the third, and so on. The
numeral was a mark made with a wedged stick pressed, into clay. The impressions were
added together from left to right.
MEMEL
/0*
The trouble with not having a zero can be brought out by the following example.
V V can represent 61, or 3660, or 3601, or 120, or even 2.
Quite often, the number depended upon the context of its use.
Suggested Pupil Activities
1. Sample seatwork exercises:
a. Compare the value of the places in the Babylonian system with those in the HinduArabic system. In which system does place value increase at the same rate?
b. Problem: Would you write 15 as V
or as
? Why? How do you suppose the Babylonians symbolized this? (Hint:
V
long, narrow, horizontal tablets were used.)
c.
d.
How many different number symbols did the Babylonians have?
What number do you think the Babylonians meant by
e.
What does the Babylonian numeral f.
g.
What plan did the Babylonians use in writing their numerals?
What number is represented by each of the Babylonian numerals below?
six hundred
civtti
one
rv
'.1
I
54
mean?
-
V'
V1
h. What number is represented by the numeral in the following sentences?
1) john and Bill are V V boys.
eggs in a dozen.
2) There are 4 V V
eggs in a gross.
3) There are VV
VV VT
minutes in an hour.
4) There are
i.
Solve:
7
2) i4YY
—+ Id
V THE MAYAN NUMERATION SYSTEM
The Mayan civilization was the oldest of all the aforementioned civilizations; yet it had
the most progressive numeration system. It not only utilized place value, but it also contained a zero symbol in somewhat of a base-20 system recorded vertically. Only three
unique symbols were used:
=0
- =5
The symbols were employed in an additive situation, which increased vertically.
a. _5/=
b.5.i-S--i-i=//
When the lines are close together, such. as ' - ', the numeral represents 10. However,
when they are separated, such as ', the numeral represents 5 in the 200 place and
5 in the 201 place, thus giving a total of (5 x 200 ) + ( 5 x 20 1 ) or 105.
Here are several Mayan 'numerals':
13
a.
d.
b. —L. 7)(.20
11K/ —/1
e.
0
I_s_I
I X.20
ao
/00
7x/
/07
f.
/ ( .2o'
/
220
/ KO
9000
I x
00
4o0
1K .2a =
?.20°
In chart form, then, the place value progressed as follows:
204
20
20 2
20'
20°
160,000
8,000
400
20
1
Suggested Pupil Activities
1. For pupils in the middle intermediate grades who are not familiar with the exponential notation, proceed by telling them that each successive place is 20 times larger than
the former. Allow pupils to arrive at the values of the places on their own.
55
2. Give pupils exercises in changing numerals from the Hindu-Arabic system to the
Mayan numeration system and vice versa.
3. Sample seatwork exercises:
a. . .. .-i
(Answer in Mayan numerals.)
(Answer in Mayan numerals.)
b.
.- ç
c.
^
(Answer in Mayan numerals.)
d. In the Mayan system, what is the greatest number of '—s' that you could write in
one place value? How many'. s' can you write in one place?
e. In what major way is this system different from the Hindu-Arabic? (Number of numerals and in reading.)
f. With an example, show how numbers could be confused in this sytem if there were
no zero symbol.
VI THE HINDU-ARABIC NUMERATION SYSTEM
It is hoped that the pupil will have a better understanding of the system we use today,
the Hindu-Arabic, and perhaps why it is more convenient than the ones outlined in this
unit.
Of interest might be how one person from Morocco more than 1 'Q00 years ago conceived
the Hindu-Arabic numerals. ,' Z 3 4 S
It X B 0
Note that the symbols contain the same number of angles as the numeral names.
In Retrospect: (Culmination exercises to tie the six numeration systems together could be
derived from the following chart.)
NUMERATION
SYSTEMS
IN
ORDER
OF
COMPLEXITY
NUMBER OF NUMERALS
0
8
C
C
0
o
0
.-
0
0
<.0
1. Zambian
1
1
1
2. Egyptian
2
3
4
5
3. Roman
3
5
7
4. Babylonian
2
2
5. Mayan
3
6. Hindu-Arabic
10
Y — yes
56
NIL
tn
0
Sog
N
0
3
,°
0
1,
i.t'co
o
.
o
E
Z .!
Q U..
Y
N
N
10
Nil
Y
Y
7
Y
N
N
10n Nil
Y
Y
7
7
Y
N
N
Nil
Y
2
2
2
Y
Y
N
N
Y
6&
10
3
3
3
3
Y
Y
Y
N
Y
10
10
10
10 EN
Y
Y
Y
Y
N—no
Alternate
20n
ion
Math Games for Intermediate Grades
by R. MELENDEZ-DUKE, Willows Elementary School, Victoria, B.C.
Summary
The arithmetic games herein described are the result of five years of experimentation.
I wished 'to make a killing' out of selling games, but this was not so easy, as patents,
copyrights, buyers and manufacturers proved to be costly and scarce. Consequently,
I'm tied to the classroom for another 20 years, as royalties from the games will never
provide bread for the table, much less early retirement.
Sincere thanks to the Greater Victoria School Board and Roy Lister, Director of Instruction, for making my time available to set down my ideas in a followable form, and
Gordon Smith and Ian Paul for their valued advice and encouragement.
The materials presented here, as well as others, have been the subject of in-service workshops for the Greater Victoria School Board and the Fourth Mathematics Summer Workshop during 1975.
These games have a grade range from 5-9, and some could be used for grades lower, and
higher. They have been particularly successful in Grade 7. Topics include:
place value
computation
evaluation
factorization
bases
It is hoped the ideas presented will be used or modified by teachers and that the ideas
will open up new avenues of thought.
Games can be great fun besides strong reinforcers of concepts learned. Games, through
the use of cards, spinners, dice, provide elements of chance, reward, punishment and
friendly competition.
In general, the fewer moving parts and 'losable' parts, the better the games are.
The materials referred to throughout are: cardboard, manila tag, and dice (2.54 cm cubes
- the polite way to say one-inch cubes); rodding material 3/4" x 3/4" x 36" (say), available
at lumber yards, will make many dice. These materials are readily available in the schools,
or they can be 'pilfered' from primary teachers. Laminating cardboard or Manila tag can
add years of use to materials that do not have the durability of wood.
Permission is given to all to use; copy any activity herein described.
57
RECIPROCITY
Skills:
Making high and low numerals from two given numbers, addition, multiplication by two, division by two —or X 3, X 7, 5, 3.
Players:
An entire class or, indeed, the entire school may participate.
Materials:
Each youngster is to make a box as indicated in the drawing.
Two dice - one having faces 0, 1, 2, 3, 4, 5, the other having faces
0,5,6,7,8,9
Spinners, bearing the same numeration, may also be used.
Procedure: Each player must decide where to place the following directions:
High, low, double high, double low, half high, half low.. . beside the rolls.
It must be assumed that youngsters will place H, L, DH, DL, ½H, ½L at
random beside each small box . . there should be 6. (factorial 6) or
6 x 5 x 4 x 3 x 2 x 1 equals 720 different ways in which to place H, L, DH,
etc.
Roll dice. Compute after each roll. Add.
Dice Values
A
.
B
C
D
3, 6
1 1 32
½H
1
126
DH
1
36
L
1
36
5,5
2 110
DH
2
itO DL
2
55
H
2
55 H
0) 8
3
08
L
3
80
H
3
04
1/2L
3
04
'AL
114
4
41
H
4
21
%H
4
21
'AH
.4
21
½H
4,7
5
24
'AL
5
47
L
5 148 DH
5
94
DL
2,9
•6
58
DL
6
15
'AL
6
6 184
DH
Totals
273
399
58
322
DL
L
394
Variation: Let the H, L, DH, etc., be fixed by the teacher, youngsters place 1, 2, 3, 4,
5, 6, as per their choice. Multiplication, division by other numerals is
possible. Round up answers as customary.
58
.
PLACE VALUE GAME
Skill:
Place Value
Materials:
Paper, die with numerals 0, 1,2,3,4,5 or 1, 2, 3,4,5,6.
Procedure: 1. Let all youngsters make a box thus:
Variation:
2.
Roll the die. The player may put that value in any box the player
wishes, but the idea of the game is to make the greatest five-digit
numeral.
3.
Roll the die five times.
4.
Example. If the following faces turned up (0, 2,
5, 3, 2) in order, the
winner would be the youngster who had 53220. Some youngsters,
through choice, would have obtained 25230.
Play for the largest and smallest five-digit numeral. In the abOve example,
the lowest numeral would be 02235.
Another Variation: Make boxes thus:
less
HI
1. Roll die seven times. After each roll, player places that value anywhere
player wishes.
2. Subtract as customary. Theplayer with the largest difference wins.
3. As a further variation, players with largest and smallest difference may
be declared winners.
Inventors: This game can be extended to other operations. Can you invent your own?
Can you use another die with different values?
59
• FACTO-T ICKY-TACKY
Skill:
Materials:
Finding the prime factors up to 99.
Cardboard, wood or Manila tag showing composite numbers. Fifty-four
cards with prime factors.
10 cards for No..2
10 cards for No.3
10 cards for No. 5
5 cards for No. 7
5 cards for No. 11
Scards for No. 13
3 cards for No. 17
3 cards for No. 19
3 cards for No. 23
Total 54 cards
8 9110 12 14 115
16 18 20 21 22 24
25 26 27 28 30 32
33 34 35 36 38 39
40 42 44 45 46 48
49 50 51 52 54 55
56 57 60 63 64 65
1
66 68 69 70 72 75
One Board
76 77 78 80 81 84
85 88 90 91 92 95
P96 9899
Procedure: 1. Two pupils.
2. Deal out five cards each.
3. Find all factors for a numeral on the board. Placetwo or three prime
cards for the numeral. One player places cards face up; the other, face
down.
4. After each play, a player replenishes to five cards by taking from the
pile as many as played. If he has not been able to play or elects not
to play, player takes one card from the pile.
5. To win, a player must have three composites in any direction filled with
cards, thus:
I
X
e
6. Blocking an opponent who has 'found' two boxes is permitted, thus:
Variation:
60
This game can be adapted to operation of addition for primary children if
'facts' go from 6 to 50.
.
EVALUATION
Quick computation, evaluation using conventional order.
Cardboard or wood, spinners, pins 3 . Ten dice.
Skill:
Materials:
Board: F
I
W30
J
0 - .0)] + [ 0 X (0
^
0)
Ten
' (0
(0-D+D)X t[ 1]-
0) 1
0
q
I 5_z7 I
Units /fl
Numerals on Faces of Dice
Die No. 1.
2.
3.
4.
17,
11,
16,
20,
5.
41
6..
7.
8.
9.
10.
5,
7,
5,
3,
51
16,
2,
10,
8,
11,
6,
8,
12,
61
14,
4,
13,
3,
7,
10,
9,
4,
01
19,
20,
0,
1,
13,
8,
01
15,
15,
14,
8,
12 1
7,
4,
17,
1,
.9,
3,
6,
'2,
5,
21
3
0
7
6
2
1
1
18
19
18
Procedure: 1. Two players take turns on the board.
2. Cast all dice, face on die indicates what numeral shall, be played.
Do
not turn dice.
3. Spin both spinners to find out what two-digit numeral must be arrived
at. . . tens and units.
4. Use top equation first. If this equation does not work out with the
numerals, attempt the bottom equation.
5. Changes may be made to the last (right-hand) box. Let the symbol be
+ or -. An unsuccessful attempt does not mean failure as the student
should have gone through a great deal of computation to conclude
numerals do not fit the evaluation.
Variation:
Ignore the spinners and board. Who in a class can come up with the
largest
or smallest answer, given certain dice values and certain symbols to be used?
61
Learning Di fficul ties in Mathematics
by DAVID F. ROB ITAILLE, Faculty of Education, UBC
John is 10 years old and Mark is 11. They were referred to the Mathematics Education
Diagnostic and Instructional Centre (MEDIC) at the University of British Columbia for
identification and treatment of learning difficulties in mathematics. Can you pinpoint the
nature of the errors (see fig. 1) they made during a diagnostic session?
JOHN
MARK
235
x7
2023
2
12
15
Figure 1
Children with learning difficulties in mathematics may be found in virtually every classroom. They are not necessarily slow learners or mentally retarded or emotionally disturbed. As a matter of fact, many of them do relatively well in all their school subjects
with the exception of mathematics. What causes such problems? How can they be identified? And, most importantly, what can teachers do about helping students overcome
such problems?
Causes of learning problems in mathematics
The majority of learning difficulties in mathematics fall into.four broad categories.
1. Gaps in knowledge. Mathematics as a subject is highly developmental in nature. In
other words, a student's progress in mathematics may be slowed or halted because of a
lack of mastery of prerequisite skills l and concepts. Thus, a student who cannot subtract
will be unable to master the division algorithm Similarly, someone who has not mastered
the basic facts for addition will not be proficient at using the addition algorithm.
2. Incorrect learnings. Many students who are referred to MEDIC have taught themselves incorrect procedures. Thus John (fig. 1) knows the basic facts for multiplication,
but he has somehow learned to write down the tens' digit and carry the units' digit rather
than vice versa
times 5 is 35. Write down 3 and carry 5'), Mark has learned that any
computation with fractions involves the calculation of cross-products ('2 times 6 is 12;
3 times 5 is 15').
3. The student's developmental level does not coincide with the teacher's presentation
0
level. Much of the teaching of mathematics takes place at a fairly abstract level. This is
unfortunate for many of the students, since they need to work at the concrete level as
often as possible. Pictures and illustrations in a textbook are good and should be utilized,
but they are not concrete enough in many cases. Children need to make use of concrete
materials that they can manipulate. A good example of this occurs in the area of placevalue concepts.
An extremely high proportion of the students referred to MEDIC have difficulties with
place-value. Frequently, this may be because such concepts are taught on a level that is
62
too abstract for the students. Teachers should make more use of place-value teaching
aids such as pocket charts, the abacus, and multi-base blocks.
4. Severe emotional or intellectual handicaps. In some cases, a student's difficulty in
learning mathematics may be symptomatic of an emotional or intellectual disorder. Research has shown that educable mentally retarded children can learn mathematics, but
that their progress is very slow (cf. Peterson, 1973). Techniques for teaching mathematics
to such children are discussed in detail in the Peterson text, Functional Mathematics for
the Mentally etarded.
The Diagnostic-Remedial Process
Teachers who are willing to devote the required time and energy can follow these steps
in working with their own students who require remedial assistance.
1. Diagnosis. The reason for conducting a diagnostic session in mathematics is to identify the precise nature and the underlying causes of a student's difficulties. Unless such
information is obtained, remediation will probably consist of treating the symptoms only.
Such a procedure will often prove to be either ineffective or inefficient.
For example, a diagnostic session with John would result in a finding that he has difficulty with place-value concepts. He knows how to use the multiplication algorithm, but he
fails to make proper use of place-value ideas in writing down the partial products. A successful remedial program for John would likely begin with a reteaching of the appropriate
place-value concepts.
There are a number of standardized diagnostic tests in mathematics available. A listing of
these may be found in the National Council of Teachers of Mathematics pamphlet 'Mathematics Tests Available in the United States.'
On the whole, such tests are useful for identifying students who are performing poorly,
but not for obtaining the kind of precise information needed to make an accurate diagnosis. Thus, administration of a diagnostic test to John might result in a finding that he cannot perform multiplication exercises correctly. As mentioned above, this information is
not precise enough to be very useful in planning a remedial program.
A second factor militating against the use of standardized diagnostic tests on an individual
student basis is their uneconomical use of time. For example, a student who has no difficulty adding or subtracting whole numbers must still respond to all of the test items dealing with these algorithms. As a result, it may take several hours spread over several testing
sessions to administer the test. Level II of the Stanford Diagnostic Arithmetic Test requires three hours and fifty-five minutes to administer in seven testing sessions.
The most efficient way to obtain a precise diagnosis is through a diagnostic interview
between the diagnostician-teacher and the student on a one-to-one basis. For such interviews at MEDIC, we make use of a locally developed checklist (see Appendix), but classroom teachers may prefer to select items from one of the available diagnostic tests. Given
an item, the student is asked to perform the task in the teacher's presence, describing
the steps taken and, wherever possible,-the reasons for so doing. In this way, the student's
reasoning will become clearer to the teacher, and the causes underlying the difficulty
will be more easily identified.
63
To make more efficient use of the student's time, the teacher can present tasks for the
student to do in descending order of difficulty. For example, in the case of addition of
whole numbers, the first item might require the student to find the sum of three or four
three-digit numbers where regrouping is required. If the student completes the task with
ease, no more items involving addition of whole numbers need be presented, and the
diagnostician may proceed to the next skill or concept. A great deal of time may be saved
in this way, but care must be taken to ensure that the student does not become frustrated
by repeated failure to perform successfully.
2.
Determine the student's algorithm-confidence level. Most of the students referred to
MEDIC display behavior similar to John's and Mark's. That is, they have developed their
own incorrect algorithms, which they use fairly consistently and extremely efficiently as
a result of extensive practice. Moreover, when asked about their methods, such students
all too frequently display a fair amount.of confidence in the correctness of their idiosyncratic algorithms.
If a student believes an incorrect algorithm to be the correct one and if the student is
adept in its use, the remediator is faced with a special problem. Simply telling such a student that the method is incorrect will not likely be sufficient to convince him. The student must somehow become convinced that his method is incorrect before remediation
can be truly successful.
3. Set objectives. Objectives for the remedial program may be specified once the diagnosis is complete. These objectives should be stated with as high a degree of specificity
as possible if they are to be useful in deciding upon the success or failure of the remedial
program. Behavioral objectives, objectives stated in terms of the anticipated student behavior, seem particularly well-suited to this purpose. For example, an objective worded
'To bring him up to a Grade 6 performance level' is difficult to interpret. It would be
preferable to replace such an objective with a series-of more specific objectives such as
'To be able to divide a whole number by a two-digit divisor.'
4.
Plan the remedial program. Some commercially available diagnostic-remedial materials contain plans for remediation. These plans typically consist of references to particular pages and sections of elementary school mathematics textbooks. Thus, if a pupil has
difficulty with subtraction, the teacher is referred to the appropriate pages in each grade
level of the textbook series being utilized. The student is then expected to work through
all of this material under the teacher's supervision.
There are at least two major flaws in such an approach. First, as an approach to remedia tion, it fails to take into account what the child already knows about the skill or concept
in question. Second, it fails to take into account the specific nature of a child's disability.
John (fig. 1) knows the multiplication algorithm with one-digit multipliers; he needs help
with place-value concepts.
A remedial program should be tailored to fit the needs of the individual student, taking
into account strengths and weaknesses. Information regarding a particular student's requirements should be collected during the diagnostic interview.
64
C'-
5. Select teaching method(s). There are a great number of possible teaching approaches
from which one can choose, and it is impossible to stipulate which method or methods
are guaranteed to be successful with all students. However, some aspects of methodology
for remedial teaching can be stated categorically. For example, it is important to make
use of concrete manipulative materials in remedial teaching at every opportunity. Also, if
a particular method of teaching a topic has been implemented, and the student is still not
succeeding, a change in method is called for. It is inefficient and wasteful of both the teacher's and the student's time to continue using the same approach after it has proved to
be. unsuccessful. Third, a method different from that used in the original teaching of the
skill or concept should be used in remedial work. If the student failed to learn by that
method in the first place, the student is likely to do so again.
6. Remediate. Plan each remedial session carefully. If possible, try to account for each
minute of each session, to make the most efficient use of the time available. Schedule a
number of short remedial sessions rather than one long one. The more frequently, you are
able to work closely with the student, the less will be the possibility, of his acquiring new
incorrect learnings or reverting to old errors and the more you will be able to provide
support and reinforcement.
Support and reinforcement are important aspects of remedial teaching. Remember that
students who are having difficulty with mathematics usually have had such problems for
some time. A history of failure may cause frustration, negative attitude toward mathematics and, perhaps, a poor self-concept. If you can provide a remedial experience that
permits the child to experience success in mathematics, some of the negative feelings
may be alleviated.
7. Evaluate. The objectives stated in point 3 above may be used as a basis for generating
test items to evalute the student's progress. Alternatively, you may want to use an achievement test for this purpose. You should not use a diagnostic test to measure achievement, since, as a general rule, items on diagnostic tests are easier than corresponding items
on achievement tests. Also, there are usually more items on a diagnostic test than is necessary for evaluating achievement.
The results of the evaluation must be acted upon. If all has gone well, the student may
proceed with the remainder of the remedial program or rejoin classmates who are working on the main-line program. If the remediation has not been successful, a recycling
through the process is required. The recyling may, in some cases, necessitate only a
change in methodology. In other, more difficult cases, a reformulation of the set of objectives and, perhaps, a lowering of the expected level of performance may be necessary.
Conclusion
Students with learning difficulties in mathematics can profit from a process of diagnosis
and remediation. John and Mark did, and so did the other students with whom we have
worked at MEDIC.
You can help your students who are experiencing difficulty in mathematics by following
the suggestions made here. Of fundamental importance to the success of this approach
are these two points: first, diagnosis in mathematics can best be done in a one-on-one
65
interview; and, second, a remedial program must be designed for the individual child,
taking into account the child's particular strengths and weaknesses.
References
Beatty, L.S., Madden, R., and Gardner, E. F. Stanford Diagnostic Arithmetic Tests, Levels
land II. New York: Harcourt Brace, Jovanovich, 1966.
Braswell, James S. Mathematics Tests Available in the United States. Reston, Virginia:
NCTM, 1972.
Peterson, Daniel. Functional Mathematics for the Mentally Retarded. Columbus, Ohio:
Merrill, 1973.
National Council of Teachers of Mathematics The Slow Learner in Mathematics (35th
Yearbook). Reston, Virginia: NCTM, 1972.
Reisman, Fredricka K. A Guide to the Diagnostic Teaching of Arithmetic. Columbus,
Ohio: Chas. E. Merrill, 1972.
ELECTION RESULTS As a result of the votes mailed in by members, the following persons
were elected to two-year terms on the BCAMT executive, effective
July 1, 1976.
VICE-PRESIDENT
TREASURER
PUBLICATIONS CHAIRPERSON :
66
WILLIAM A. DALE
MRS. GRACE DILLEY
MRS. SUE HABERGER
METRIC MEASURE
by JANE SRIVASTAVA, North Vancouver, B.C.
[Editor's Note: B.C. schools have recently been supplied with copies of a publication,
Metric Style Guide, from the Council of Ministers of Education (see bibliography). This is
an attractively laid out resource book for teachers. In this article, Jane Srivastava outlines
some important details of metric measure and shares some classroom learning acitivities.]
METRIC MEASURE: What is it? Why all the fuss?
The federal government has committed Canada to a change to the metric system, to be
completed by 1981. In B.C., primary teachers were instructed to begin teaching the
metric system of measurement in the fall of 1973; intermediate teachers, in the fall-of
1974. Publishers of textbooks and distributors of teaching materials inundated the mar-.
ket with new metric materials - the hottest items being conversion charts - and the local
newspaper prominently displayed a photograph of a classroom in which the teacher had
written on the board 'Inches are now centimetres.' Inches are no more centimetres than
they are feet or yards. The teacher probably meant to say that we now measure in centimetres instead of inches. Watergate, the energy crisis, the Mideast war faded into insignificance as teacher joined Henny Penny in declaring 'the sky is falling' and felt inadequate
to measure how much of it was falling, or how far.
One can imagine the same chaos, confusion and bitter debate when the Arabic system
replaced Roman numerals: in both cases, the new system is more efficient and consistent; the old has only tradition to recommend it.
The history of measurement, of man's progression from non-standard to standard units,
is well-known. The adoption of the metric system, and the standardization of notation
in that system, is but the latest step in our progress toward a coherent and universal
measuring system.
How do we gain?
One of the most convenient aspects of the metric system is its organization on the decimal system: a centimetre is 0.01 metres; a kilometre is 1000 metres; a kilogram is 1000
grams. How nice to be able to reinforce the teaching of place value in our measurement
activities as well as in computation of money amounts - no more long hours of teaching how to carry from the ounces to the pounds column or from the inches to the foot
to the yard column. J ust move a few decimal points and the work is done. Computation
with decimal fractions can take precedence over complicated fifths, quarters and thirds.
We can spend less time learning how to record and compute, and more time measuring.
Also, for a few years, we have a beautiful opportunity really to teach estimation: here is
a centimetre; try to find things that are about five centimetres long, about ten centimetres long, about a metre long. The situation is not artificial, because most of the things
in our classrooms and homes were measured in inches and feet when they were manufac-
67
tured, and they cannot be measured in even centimetres and metres. Because metric units
are relatively new in our experience, we get good experience in looking at, feeling and
visualizing new quantities. This situation won't last long, we must take advantage of it
now.
What do we teach?
We have a new word in our vocabulary: mass. Mass is constant and refers to the amount
of material an object contains. Weight, in contrast, changes. Weight is a measure of the
force of gravity exerted on the mass of an object. When we weigh an object on a balance
to- determine its mass we use standard weights whose mass has been calibrated with the
standard for mass, the international prototype of the kilogram. Although the numerical
difference between mass and weight is insignificant at most places on the earth's surface,
the concepts are distinct.
If we use these terms correctly with children, they will learn them easily. It is only because we have confused weight with mass for such a long time that the concept of mass
seems complicated to us.
Pronunciation of metric units is easy if one just remembers these guidelines: stress the
first syllable of the prefix for metric units, the second syllable for measuring instruments.
An 'o dora e ter' measures kil o me tres.
Metre, litre and gram are spelled as in this article. Other spellings seem more logical to
many people, but these are the spellings recommended by the Canadian. Metric Comm ission, and the spellings used in the official translation of the International Bureau of
Weights and Measures documents. Many U.S. companies have adopted this spelling in
;instruction manuals and sales literature. Since one of the reasons we are adopting the
metric system is to move in the direction of international standards and usage, it seems
wise to adopt internationally approved spelling. In any case, it is far more important to
know what a metre is and how to measure it than to worry about how to spell it.
Most units are written with lower case letters. The degree Celsius is an exception because it was named after an 18th century astronomer. Symbols are also lower case, and
are not followed by a period or, when plural, by an 5: I m (1 metre), 36.m (36 metres),
45 kg (45 kilograms). Units of area and volume are written with exponents:
5 km2
(5 square kilometres), 3 m 3 (3 cubic metres), 52 cm 3 (52 cubic centimetres). The symbol
for litre is a script X to avoid confusion with the numeral I. The symbol for millilitre is ml,
since there will be no confusion in this case.
The International Bureau of Weights and Measures uses spaces instead of commas to indicate place value (3 867 421. 0982) and always writes a 0 before the decimal point for
numbers less than one (0.95). 1000 is conventionally written without a space after the 1.
One should regard these details of vocabularly, pronunciation and spelling as one does the
arithmetic facts: items to be understood, then memorized - items the memorization of
which allows us to better enjoy the investigation of the concepts of measurement.
68
How do we teach it?
Why teach measuring in the metric system any differently from the system currently in
use? When we teach children about base 2 or base 5, the laws of arithmetic don't change,
just the names we give different quantities. When we teach children measurement in the
metric system we merely choose different quantities and, of course, different names as
units of measurement. Let's not confuse the issue by converting back and forth between
the two systems. Why worry about what part of an inch a centimetre is, or how many
grams in an ounce? Inches and ounces will soon be part of our history lessons instead of
our math classes.
Here are some activities to try with students:
Mass
Using a commercial set of metric weights as a reference, find stones that weigh 25 g,
50 g, 100 g, 500 g, 1 kg. Choose the stones by heft alone, then check your accuracy on
a balance. Make your own set of weights out of clay. Take a side trip into history: what
shapes were used for the first sets of weights, what materials and what shapes are the
most practical to use for a set of weights? What weights should you include in your set
to weigh all masses from 25 g to 1 kg?
Length
Measure off a kilometre by taping together 1000 metre lengths of string. Make different
paths with the metre string on the playground. With the permission of the appropriate
authorities, measure and paint a kilometre length on the sidewalk or road near your
school. Erect a sign at the beginning and end of the kilometre. Time yourselves running
or walking a kilometre.
Area and Volume
Use cardboard to build square and cubic centimetres, decimetres, metres. One of the
early definitions of a kilogram was the weight of a cubic decimetre of water at a certain
temperature. Check this by finding the mass of 1 dcm 3 of water.
Mass and Capacity
Collect boxes and bottles (toothpaste, cereal, contact lens solution) that give mass or
capacity in metric units. Display them in your classroom or hall corridor. Note how much
easier comparison shopping will be using the decimalized metric system!
Try these recipes:
1. Litres of Lemonade
Mix together in a large jar or bucket:
1
2 water
350 g sugar
the juice from 3 lemons
(for extra flavor, put the lemon peels in, too).
2. Metric Marsh Krispies
Melt 50 g butter or margarine in a double boiler.
Add 300 g marshmallows.
Cook until the marshmallows are melted and mix well with the butter.
Add 1 50 g Rice Krispies.
Mix together well. Press mixture into buttered pan. Chill until firm and cut into 5 cm
squares. Makes about 25, depending on size of pan.
69
3. Kilograms of Clay
Sift together:
550 g salt and
650 g flour
Mix together:
500 ml water and
100 ml cooking oil
Add a few drops of food coloring.
Mix the flour and salt into the liquid, and knead until smooth. Add a little more flour if
the mixture is too wet. Add a little more water if the mixture is too dry.
Keep the clay soft in a plastic bag in the regrigerator. Let it harden by leaving it open to
the air at room temperature.
A Short Bibliography
Metric Style Guide, Council of Ministers of Education, 252 Bloor Street West, Suite
S500, Toronto, Ontario.
Curriculum Resources Board, Department of Education, Parliament Buildings, Victoria.
Canadian Metric Association, P.O. Box 35, Fonthill, Ontario LOS 1 EO. Annual membership $5.
70
COMPUTING SCIENCE
Computer Science
at ChurchillSecondary
by JAMES NAKAMOTO, Sir Winston Churchill Secondary School, Vancouver, B.C.
As schools and school districts acquire the use of computers to offer related courses at
the secondary school level, a very careful analysis must be made as to the direction and
development of the curriculum. The following is presented as a working example of a
comprehensive secondary school computer science program. Schools and districts planning to introduce computer science in their curriculum or striving to modify an existing
curriculum will, we hope, be aided by this report on our five-year growth.
To describe the development of the computer science curriculum at Churchill Secondary
as explicitly as possible, I shall comment on three aspects relevant to the evolvement of
the program. They are:
I. Chronology
II. Description of the present program
Ill. General observations (things we've learned)
I. Chronology
Computer Science and computer-oriented mathematics were introduced to Churchill
students in 1971 by J . Schellenberg, the mathematics department head. He singlehandledly pursued the project with vigor and determination for the first three years.
From the modest beginnings of a mere computer club in the first year, the project has
grown, in the short space of five years, to a large and somewhat sophisticated program.
In chronological order, the program grew like this:
1971 - One class of Grade 8s timetabled for enrichment work including some programming in BASIC. Small units introduced in other classes, not including Math 12. Com puter club organized.
1972 - All Grade 8s given a choice to enroll in 'enriched' 8X class; one class of 8X timetabled for 'computer oriented', math. One class of 32 enrolled in CS1 - included
students Grades 8-12.
1973 - One Computer Science class of Grades 9-12 included CS1 and CS2 students. Two
8X classes organized on a student selection basis; one 9X class kept relatively intact
from previous year.
1974 - Two CS1 (beginner) classes, Grades 9-12. One CS2 (advanced) class. Two 8X
classes; two 9X classes; one lox. Each instructor added small units in Math classes,
9-11.
1975 - Three CS1 classes of 30 each; two CS2, CS3 classes of 20 each. Five 8)( classes;
two Ma9x classes. Short units presented in all math classes.
1976 - Projected plans are for four CS1 classes, Grades 9-12; two CS2 classes; one CS3
class. Five Ma8X classes; two Ma9X classes.
71
II. Description of Our Present Program
Our school enrolls Grades 8 through 12; the present enrollment is 1,600. At the present
time, the following courses and projects comprise our computer science program:
A. Computer Programming Courses
1. First-year Computer Science ('CSI ' - 3 classes)
2. Second-year Computer Science ('C52' - 2 classes)
3. Third-year Computer Science ('CS3' - 1 class)
B. Enrichment Mathematics Courses
1. Math 8X - (4 classes)
2. Math 9X - (2 classes)
C. Computer Projects
1. University of B.C./Churchill Interaction Groups
2. Churchill/Grade 7 (Feeder Schools) Interaction Groups
The following is a brief description of course and project content:
A. Computer Programming Courses
The following courses are project oriented. The students are encouraged to proceed at
their own pace. The instructor attempts to provide a maximum of individual counselling
and instruction. Guidelines and short-term goals for students ensure a certain minimum
of material accomplished. Texts and references used are the CS I, CS II, CS III, and CS IV
modules by Petrak and Schellenberg' and voluminous resource books as compiled by instructors.
1. CS] - In this first course in computing science, the students are taught the fundamentals of BASIC programming - language syntax, documentation; students are encouraged to experiment and to explore new ideas and methods.
2. CS 2 - In the second year, the students are instructed in the use of logicals; they
pursue an intensive study of modularized programming - using external subroutines and
data storage. Projects accomplished to date are in these two categories:
a. interactive games - including NIM, BLACKJACK, BATTLESHIP and CHECKERS
have been student-written at our school.
b. files management - simulated airline reservation system, simulated employee master
files, and stock market predictions are examples of student-written programs.
3. CS 3 - Social and commercial applications are dealt with in this third-year course.
Topics include: (a) data base simulations and applications, (b) masking disc files and
(c) the use of simulated COBOL and ASSEMBLER languages, with the view that such
experience is seldom offered elsewhere. This course has been developed at our school
and has been approved as a locally-developed course for the coming school year.
J. Petrak and J. Schellenberg, Mathematics in Modules: CS I, CS II, CS I/I, CS IV,
Rand McNally.
72
B. Enrichment Mathematics Courses
The use of computers complements and supplements mathematics topics very well. Algorithmic thinking involved in the solution of mathematical equations, simulations of sequences, estimations of irrationals, as well as the computation of tedious arithmetic, to
name a few, all are exercises that can be translated into computer problems.
At the Grade 8 and 9 levels, these applications can be more functionally dealt with from
the standpoint of availability of time. At the senior level, the math curriculum always
seems too crowded for these lateral applications.
1. Ma8X - The regular Math 8 course is complemented by some computer programming work. Topics include the use of formulas and probability demonstrations.
2. Ma9X - The regular Math 9 course is complemented by some computer programming work. Topics include exponentiation and graphing.
Topical outlines of our programs in these grade levels are avilable on request from the
Churchill mathematics department.
C. Special Projects
1. Through interaction with UBC Faculty of Education students, our math department
has organized student computer clubs sponsored by Ed 491 college students. This provides Churchill students an opportunity to learn another language. The experience exposes students to other computer languages as well as to UBC computer installations. It
also provides UBC students valuable teaching experience.
2. Each year the math department makes a special effort to familiarize the Grade 7 students from our feeder schools, with some elementary aspects of computer programming.
This project, now in its fourth year, introduces Grade 7 students to optic card marking,
computer art and our Ma8X course. It outlines the options available to them of selecting
a computer oriented math or a regular math class. Other benefits accruing from this project are obvious.
Ill. General Observations Regarding the Development of Computer Science and
Computer Related Math Curricula
A. On the Course
1. Considerable publicity and careful preparation and planned recruitment of students
are prerequisite to a successful launching of a computer project in a school.
2. The optic card/batching mode appears to be the only means of survival for the classroom teacher. The optic card reader is a MUST if expansion of the program is an objective.
3. Efficient computer use will require the establishment of schedules and sign-ups.
4. Out-of-school hours will need to be allocated to deserving students.
5. User IDs and passwords must be kept secret.
6. As offering CAI is NOT the intent of the program, student interaction with the computer on a one-to-one basis is expensive and restrictive on numbers.
73
B. On the instructor
1. No teacher should teach any more than one computer programming course. The
individual attention they must provide each member of the class is most demanding on
their time.
2. Before teaching a programming course, the instructor is advised to teach a short
computer unit in their subject area.
C. On student work
1. Allow for varying levels of difficulty with each problem assigned.
2. Insistence upon program documentation is essential.
FEBRUARY 1976
INSTRUCTIONAL SERVICES CIRCULAR
DEPARTMENT OF EDUCATION:
843 Course Changes 1976-77
The following preliminary information on
some course changes is provided to assist in
planning for the 1976-77 school year.
Junior Secondary School Mathematics:
The fact that the revised Mathematics 9
and 10 courses were introduced on a twoyear permissive basis in the 1975-76 school
year has raised some concerns at the senior
secondary school level. These concerns
center on course planning for entering students, some of whom may have taken the
new courses and some of whom may have
taken the 'old' courses. To assist in limiting
this problem to one year, the Department
would encourage schools that have not yet
implemented the revised courses to do so
in the 1976-77 school year.
Senior Secondary Mathematics:
At the present time, work is proceeding on
finalizing course prescriptions for new
mathematics courses at the senior secondary school level. Preliminary information
on these courses is included in the curriculum guide: Mathematics Years 1-10 and
Preliminary Draft for Years 11 and 12,
1975.
a. Algebra 11 (Alg 11): This course is
identified in the 1975 guide as 'ATAG' 11.
b. Algebra 12 (AIg 12): This course, identified in the 1975 guide as 'ATAG' 12 will
74
only be available in the second semester of
the 1976-77 school year for students who
have taken Algebra 11 in the first semester.
Notes: - Appropriate action will be taken
to ensure that students taking Algebra 12
in the second semester of the 1976-77
school year will not be denied the opportunity to write a scholarship examination.
- Algebra 11 and 12 are being prepared in
both regular and honors formats.
c. Consumer Mathematics 11 (ConMa 11)
This course is identified in the 1975 guide
as Business and Consumer Mathematics.
d. Trades Mathematics 11 (TMa 11): This
course is identified in the 1975 guide as
Industrial and Trades Mathematics.
In addition to these courses, additional
mathematics courses at the 11, 12 level are
being planned for implementation in future
years. These include Probability and Statistics 12 (Finite Mathematics), Geometry 12
and Computing Science 11.
In summary, then, for the 1976-77 school
year, schools will have the option of introducing the new courses Algebra 11
(and Algebra 12 in the second semester),
Consumer Mathematics 11 and Trades
Mathematics 11 or continuing to offer
the presently prescribed courses Mathematics 11, Mathematics 12 and General
Mathematics 11.
More Problems for
Computer Science Students
by DOUG INGLIS, Caledonia Senior Secondary School, Terrace, B.C.
[Editor's Note: These problems are taken from a longer paper entitled 'Mathematics
and the Computer.' This is the second set of problems published from this paper. ]
Linear Equations
In the present secondary mathematics curriculum, there is a great deal of material concerning linear equations. The learning of much of this material can be reinforced by the
judicious use of the computer. This section will develop several related exercises thatwill
help the students to learn many of the concepts concerned with linear equations.
linear equation in two variables is an equation that can be put in the form
ax+by+c0, where a or b is not zero. Consider 3x+4y+2O. A solution can be found by
replacing x with a value, then solving the equation for y. A more direct way to find solutions is to solve the equation for y in terms of x. For the previous equation, we have
—2-3x
4 We can replace x with several real numbers and find the corresponding values of y.
Exercise 1: A
Write a computer program to find all the solutions of 12x - 18y + 4 0 such that
Exercise 2: The slope of a nonvertical line is defined to be the number of units of vertical
change divided by the number of units of horizontal change. Using Figure 1, this can be
stated as
/A:Y-.z 1 Y.')
sloe
p
y2—y1
II
I
x 2 - xl
Figure 1
Write a computer program to calculate the slope of the line connecting each of the following pairs of points
(-2 ) -5) (3,6)
C.
a. (6,1½), (12,0)
(2,3) (1,6)
d.
(1,7)
(-2,5)
b.
75
Exercise 3: Graph each of the lines described in the previous exercise. Tell whether each
line slants upward to the right or downward to the right. What can you state regarding the
direction of the line and the slope? Extend your program so that a statement regarding
the slant of the line will be printed also.
Note: The student must now think about what the different values of the slope imply.
Often, they simply use the formula to 'crank-out' the answer. This type of program helps
them to understand the concept and meaning of slope.
Exercise 4: A horizontal line has a slope of 0. Consider, for example, the points (2,3) and
(6,3) on the line y3. Substituting these values into the slope formula, we find that
Applying the definition of slope to vertical lines, we see that they have no slope. For
example, substituting the points (6,5) and (6,10) from x6 into the formula, we get
which has no meaning.
In your previous program, include now the possibility of having points from horizontal
or vertical lines. Have the computer print out 'SLOPE IS 0. LINE IS HORIZONTAL' or
'NO SLOPE. LINE IS VERTICAL.'
Exercise 5: The graph of —4x+2y+60 intersects the y-axis at the point (0,-3); this point
is called the y-intercept of the line. The x-intercept is the point where the line intersects
the x-axis; the y-co-ordinate is, of course, 0. Write a program to calculate the x-intercept
and y-intercept of the equations y4x-6 and 2x+3y5.
Exercise 6: Extend the previous program to have the print-out include the slope of each
equation and the direction that each line slants.
Note: The student is beginning to relate several different concepts to each other. He must
understand how to take an equation, find two points and then calculate the slope. 'With
proper quidance, many students will be able to see that the slope is part of the equation
(ymx+b). This program will also help many students to understand that the intercepts
are points.'
Exercise 7: Any linear equation ax+by+c=0 can be rewritten in the form y-mx+b, where
m is the slope and b is the y-intercept. Write a computer program to print the equation
of the line when you are given the slope and the y-intercept.
Note: The teacher will have to show most students how to rewrite ax+by+c0 into the
form ymx+b.
Exercise 8: When the slope and one point of a line is known, the equation of the line can
be calculated. Consider the line with slope 4 and containing the point (3,5). Then
5 = 4:3+b
and
b=-7
Thus the equation is y=4x-7.
Modify the previous computer program so that, given the slope and one point, the equation can be printed by the computer.
76
Exercise 9: Often, all that is known about a line is two points that the line passes through.
However, the equation can be found by first calculating the slope. Write a computer program to find the equation of a line, given any two points. Also have the slope, the direction of the line, the x-intercept and the y-intercept printed out.
Note: This program will enable the students to relate all the ideas concerning linear equations to one another. When the program is completed, the student will understand how to
find all the characteristics of a straight line, given any two points.
Exercise 10: As you have already found, the equations of horizontal and vertical lines are
of a special form with characteristics different from other equations. Extend the previous
program so that similar information concerning horizontal and vertical lines is also found
and printed out.
Exercise 11: In addition to having factual information about the equation of a line, it is
often helpful to have a graph of the line. In your computer program, include the option
of having a graph printed if one is desired by the operator.
Note: Many students will be unable to write programs for Exercises 10 and 11. To write
the programs, the student must have an excellent understanding of the concepts involved
and better-than-average programming skill. However, most students should be encouraged
to attempt the programs.
Supplementary Problems
1. Solve a two-by-two system of equations.
2. Solve an n x n system of equations using the Gaussian Elimination Method.
3. Find the union (or intersection) of two sets.
4. Given three parts of a triangle, calculate the remaining parts. If it is impossible, print
a message to that effect.
5. Find the square root of a positive number by an iterative process. Calculate the
answer to three decimal places.
6. Find the factorial of any number n, n I.
7. Find the number of combinations of n things taken m at a time.
8. Calculate sin x using a series formula.
9. Given a number n, write a computer program to find the absolute value of n.
10. Write a program that will add and multiply any two complex numbers.
11. Find the mid-point of a segment joining two points in a co-ordinate system.
12. Write a program that will find the length of the curved line segment which belongs to
the graph of y x 2 between x 0 and x = 1.
13. Calculate the value of using one of the many series formulae available.
14. Given the quadratic equation y = ax + bx + c, find the co-ordinates of the vertex.
15. Write a program that will evaluate a third order determinant.
16. Write a program that will find the inverse of a two-by-two matrix.
77
MINICALCULATORS IN OUR SCHOOLS 1975
by Joseph R. Caravella
We have just entered the Age of the Minicalculator. Pocket calculators are now appearing in our society with a frequency approaching that of the pocket transistor
radio. The price of the basic minicalculator
has dropped below $10. And, the National
Council of Teachers of Mathematics
(NCTM) continues to endorse the minicalculator. as a valuable instructional aid for
mathematics education and to recommend
the use of the minicalculator in the classroom.
'With the decrease in cost of the minicalculator, its accessibility to students at
all levels is increasing rapidly. Mathematics
teachers should recognize the potential
contribution of this calculator as a valuable
instructional aid. In the classroom, the
minicalculator should be used in imaginative ways to reinforce learning and to motivate the learner as he becomes proficient in
mathematics.'
The position statement above, adopted bI
the NCTM Board of Directors in September. 1974, is still relevant today. At its
September 1975 meeting, the NCTM
Board of Directors approved a report from
the Council's Instructional Affairs Committee that identified nine ways the minicalculator can be used in the classroom:
1. To encourage students to be inquisitive and creative as they experiment with
mathematical ideas.
2. To assist the individual to become a
wiser consumer.
78
3. To reinforce the learning of the basic
number facts and properties in addition,
subtraction, multiplication and division.
4. To develop the understanding of computational algorithms by repeated operations.
5. To serve as a flexible 'answer key' to
verify the results of computation.
6. To promote student independence in
problem-solving.
7. To solve problems that previously have
been too time-consuming or impractical to
be done with paper and pencil.
8. To formulate generalizations from patterns of numbers that are displayed.
9. To decrease the time needed to solve
d ifficultcomputations.
In an article appearing in the current issue
of Today's Education, published by the
National Education Assoc., entitled 'A Calculator in Their Hands. . .The Minicalculator in Our Schools,' Dr. E. Glenadine Gibb
the president of the NCTM, states that:
'Creative use of minicalculators after the
mathematical- understandings have been
extracted will establish the minicalculator
as a valuable asset among the collections of
instructional devices already found in today's mathematics classroom.'
The NCTM, through its Instructional Affairs Committee, its conventions, its affiliated groups and its official journals, the
Mathematics Teacher and the Arithmetic
Teacher, will continue to identify and
share imaginative ways of working with
minicalculators in the mathematics classroom.
Calculator Games
collected by JOHN A. PETRAK, Windermere Secondary School, Vancouver, B.C.
1. Take the number 3025 and split it into two parts, 30 and 25. Add the two parts and
square the sum (30 + 25) 2 . What do you get?
2. Flip-Flop Numbers: Certain numbers can be 'turned around' by playing around with
their digits. Enter the number; multiply by its last digit; divide by its first digit.
Example: Use the number 2178. 2178 x 8 2 = 8712
Try these numbers: 1089, 3267, 8712, 4356 (and of course the reverse of these numbers).
Can you find any more?
3. Guess the number (six digits or less)
a. Hand your calculator to a friend and ask him to enter a number of his choice
(not more than six digits) and then multiply by 2; add 4; multiply by 5; add 12; multiply
by 10; press the = key.
b. Take the calculator back from your friend and subtract 320. The result will end
in one or more zeros. Drop the zeros, and you'll have the number your friend started with.
Example: Choose 65219
1) 65219 x 2+4 x 5 + 12 x 10=6522220
2) 6522220 - 320 = 6521900; discard the two trailing zeros, and you have 65219, the
number your friend started with.
4. Change in your pocket
a. Ask a friend to take all the change out of his pocket and count it (in cents)
Then hand him your calculator and ask him to enter the amount; multiply it by 10;add 1;
multiply by 2; add 21; press the = key.
b. Take the calculator back and multiply by 5. The result will be a number ending
in 15. Discard the 15 and subtract 1 from what is left. The answer will be the amount of
change your friend has.
Example: 43 cents change
1) 43x10+1x2+21883
2) Take 883 x 5 = 4415; discard the 15 to get 44; subtract 1 to get 43 cents. Yourfri'end
had 43 cents in his pocket.
5. What's your age?
a. Hand a friend your calculator. Ask him to enter his age; subtract his favorite onedigit number; multiply by 9; add his age; press the = key.
b. Take the calculator back from your friend. Add the two digits of the number. to
'guess' your friend's age. (If the number has three digits,.add the last digit to the first two.
digits.)
Example: Your friend is 15 and has 7 as his favorite one-digit number.
1) 15-7x9+1587
2) 8 + 7 = 15, the age of your friend.
Example: Your grandma is 67 and has 3 as a favorite one-digit number.
1) 67-3x9+67643
2) 64+3=67
79
6.
The day and month of your birthday and your age
a. Hand somebody your calculator and ask him to enter the number representing
the month of his birthday; multiply it by 100; add his date of birth; multiply by 2; add 9;
multiply by 5; add 8; multiply by 10; subtract 422; add his age; then press the = key.
b. Take the calculator back and subtract 108. The result will be a five or six digit
number. The first digit(s) tell the month of birth; the next two the date of birth; and the
last two, the age of your friend. Read them out to your friend. He should be impressed.
Example: Your friend is 15 years old and was born on March 13, 1960.
1) 3x100+13x2+9x5+8x 10-422+15=31423
2) Take 31423-108 = 31315 or 3/13/15
Your friend was born in the third month (March), on the 13th day, and is 15 years old.
7. Guessing the date of somebody's birthday
a. Hand a friend your calculator and ask him toenter the number of the month in
which he was born; multiply by 100; add the day of the month of his birth; multiply by
2; add 8; multiply by 5; add 4; multiply by 10; add 4; add the last two digits of the year
of his birth; finally subtract 3333 and press the = key.
b. Take the calculator back from your friend and add 2889. The result will be a
five- or six-digit number. The first digit(s) tell the month of birth; the second two, the
day of birth; and the last two, the year of birth.
Example: Your friend was born on August 19, 1961.
1) 8x.100+19x2+8x5+4x10+4+61 —3333=79072
2) Take 79072 + 2889 = 81961 or 8/19/61
Your friend was born in the 8th month (August), on the 19th day, of the year 1961.
8. Consider these numbers
.
a.. 153 (three digits); 1 3 + 5 3 +3 3 =
b. 8208 (four digits); 8 4 +2 4+04 +8 4=
c. 54748 (five digits); 55 + 45 +7 5 +4 5 +8 5 =
d. 548834 (Six digits);56+46+86+86+36+46=
e. 1741725 (seven digits); 1 +7 7 +4 7 + 17 +7 7 + 2 + 7 =
f. 24678050 (eight digits); 28 +4 8 + 68 +7 8 +88 + 0 8 + 8 +08 =
g. 146511208 (nine digits); i +49 +6 + 59 + 1 .9 + i +2
+8
h. 4679307774 (ten digits);4 10 +6 +710+910+310 +Q 10 +7 10 +7 10 ±7 10 4 10 =
Note: Some calculators will not be able to handle the nine-digit and ten-digit numbers in
questions g. and h.
Did you find the pattern? Excluding one-digit numbers, only 15 additional numbers have
been found (up to 10 digits long) that have the same properties displayed by the above
example. Can you find any of them?
80
Electronic Calculator Word Games
If you own an electronic calculator, here are a few nonmathematical 'fun exercises' you
may wish to try. Unfortunately the vocabulary for calculator word games is limited to
I.E.h,S,L,B 4 O,G, and D, depending upon the calculator and on your imagination.
Here are a few exercises that you may wish to try:
1. Enter 0.7734
2. Enter 77345
3. Well-trained dogs will 7334 when they are told to.
4. Complete happiness is 121 x 57 + 0.25 x 16 ±2
5. If 2,366,851 Canadians watch Hockey Night in Canada (enter 2366851) for all three
periods (enter x 3), who is the winner?
6. If 142 Arabs fight with 154 Turks over 69 litres of oil (enter 14215469) for five
years (enter x 5), who will get the oil?
7. The Arabs may 0.1283 x 3 + 47 x 15 to 71 x 2 + 0.15469 x 5 if the price is right.
2 + 2 + 2) x 0.03
8. The stock market is dropping! (508 2 -16
9. A familiar computer principle. (0.844561)
10. If you buy 100,000 shares of stock (enter 100000) at $148.18 a share (enter 'x
148.18) and pay $472 commission (enter +472) and the price goes down 25% (enter x
0.25), what do you find yourself in?
11. If you invest $211,843 in the stock market (enter 211843) and your stock falls
26% (enter x 26), what do you have?
12. We hope that these word games gave you lots of 5379919!
Answers:
1. hELLO
2. ShELL
3. hEEL
4. BLISS
5. ESSOOIL
6. ShELLOIL
7. SELLOIL&ShELLOIL
8. SOSELL
9. GIGO (Garbage In Garbage Out)
10. BIGhOLE
11. BIGLOSS
12. GIGGLES
81
List of Publisher's Representadues
U
I
compiled by BILL KOKOSKIN
[Editor's Note: This list was originally p ublished In the June 1975
Vector, It is reprinted here for the convenience of teachers in these times of curriculum revision and
multi-text course options. ]
Tony Vander Woude
Addison-Wesley Canada Ltd.
444 Robson Street
Vancouver, B.C. V613 2135 685-3622
Brian Sims
B.C. Heath Canada Ltd.
4283 Arthur Drive
Delta, B.C. V4K 2X1
946-6715
John Walker
Visual Education
1 869 West Georgia Street
Vancouver, B.C. V6G 2W2
682.3788
Peter Clark
Random House
15772 Goggs Avenue
White Rock, B.C. V413 2N7
536-9170
Chandler Hill
Denoyer-Geppert
2808 Tilden Street West
Seattle, Washington 98199
283-5071
Doug Adams - Stu Calder
Collier MacMillan Canada Ltd.
P.O. Box 8780, Station H
Vancouver, B.C. V613 4E2
Robert Wood
Ginn & Company
8566 Terrace Drive
North Delta, B.C. V4C 3Z2
581-8153
Jeff Stone
MacMillan Co. of Canada Ltd.
130 - 19th Street West, Suite 102
North Vancouver, B.C. V6M 1W4
980-2328
Lowell Smith
Thomas Nelson & Sons
1007 -1651 Harwood Street
Vancouver, B.C. V6G 1 Y2
687-6394
Adrian Peetoom
Scholastic Publications
5940 Canada Way
Burnaby, B.C. V5E 3V7
526-1414
Raoul Bury
McIntyre Educational Media 14431 Saturna Drive
White Rock, B.C. V413 3B6
531-7434
George Clarke - Paul Jones
McGraw-Hill Ryerson Ltd.
1652 West 7th Avenue
Vancouver, B.C. V6J 155
731-8013
82
Rick Pawliuk
McGraw-Hill Ryerson (EDL Div)
104 - 11460 40th Avenue
Edmonton, Alberta T6j 0R5
434-0920
Sucha Gill
Setsco Educational Ltd.
1 315 West 71 St Avenue
Vancouver, B.C. V6P 3B3
261-1361
Dave Durbin
Holt, Rinehart & Winston (Canada) Ltd. 11316 - 78th Avenue
Delta, B.C. V4C 1N7
594-6044
Wes Edwards
Gage Publishing Ltd.
600 Westview Centre
North Vancouver, B.C. V7N 3X3
985-9367
Martin Goldberg
McClelland & Stewart Ltd.
27592— 110th Avenue, R.R. 1
Maple Ridge, B.C. V2X 7E6
526-1418
Chris Thomas
Charles Merrill Canada Ltd.
1102 —1651 Harwood Street
Vancouver, B.C. V6G 1 Y2
681-1847
John Morrow
Spectrum Educational Supplies Ltd. 315- 1152 Mainland Street
Vancouver, B.C. V6B 2T9
688-6522
Cam Millar
Doubleday Canada Ltd.
202 -828 Gilford Street
Vancouver, B.C. V6G 2N6
682-0239
Hank Luck
Fitzhenry & Whiteside Ltd.
Box 3287, MPO,
Vancouver, B.C. V6B 3X8
874-0511
Alan J. MacFarlane
Dent
1510 Bowser Avenue
North Vancouver, B.C. V7P 2Y3
987-8504
Bob Minor
Readers Digest (Canada) Ltd. 3811 West 4th Avenue
Vancouver, B.C. V6R 1P8 224-5893
Mike Wood
GLC
505 Spadina Street
Chilliwack, B.C. V2P 1V3
792-8866
Bob Skinner
Educational Progress Co. 7265 Willingdon Avenue
Burnaby, B.C. V5j 3R8
433-2127
Jack Hughes
J.A. Hughes & Associates Ltd.
677 East 5th Avenue
Vancouver, B.C. V5T 1H8
872-7217
S
Trevor Fox
Van Nostrand Reinhold Ltd.
1595 East 57th Avenue
Vancouver, B.C. VSP 2B2
327-0816
.
Randy Koenig
TalensC.A.C. Ltd.
1030 Leroi Street
Vancouver, B.C. V5K 4G2
255-5378
83
Robert Blaney Ltd.
144 West 5th Avenue
Vancouver, B.C. V5Y 1 H9
872-8671
Steve Taggart
Copp Clark Publishing
2918 Mathers Avenue
West Vancouver, B.C. V7V 2K1
926-6907
Cameron Varcoe
International Tele-film Enterprises
127 West 22nd Avenue
Vancouver, B.C. V5Y 2G2
874-1436
Howard Greaves
Ed u-Med ja
5 - 27090 32nd Street
Aldergrove, B.C. VOX lAO
856-4289
Don Greenwood
Science Research Assoc. (Canada) Ltd.
403 Newdale Court
North Vancouver, B.C. V7N 31-13
985-7680
Chris Jackson
Oxford University Press
304 - 4824 Georgia Street
Delta, B.C. V4K 2S9
946-2047
84
•
• Arthur Colbert
Book Society of Canada Ltd.
5687 Wallace Street
Vancouver, B.C. V6N 2A1
266-5138
Pat McGoey
Houghton Mifflin Co.
1004 - 9500 Erickson Drive
Burnaby, B.C. V3J 1M8
939-2551
Allan Stuart
Sharp's Theatre Supplies
444 Coldfall Road
Richmond, B.C. V7C 1 P8
688-4832
Nonresident
Philip Lorenz
Longmans (Canada) Ltd.
55 Barber Greene Road
Don Mills, Ontario M3C 2A1
(416) 444-7331
Ms. Pat Foran
Psycan Ltd.
255 Consumers Road
Willowdale, Ontario M2J 1 R3
(416) 491-4343
Fi fth Mathematics Summer Workshop
by DOUG OWENS, UBC
The Fifth Mathematics Summer Workshop will be held on Tuesday, August 31 and Wednesday, September 1, 1976, at Carson Graham Secondary School in North Vancouver.
The Mathematics Summer Workshop is sponsored annually by the B.C. Association of
Mathematics Teachers and the National Council of Teachers of Mathematics.
Program:
The keynote address will be given this year by Dr. Eric MacPherson of the University of
Manitoba. He will speak on 'Curriculum Development: For Richer, For Poorer, For
Better, For Worse.' The workshop features approximately 70 workshop sessions at all
levels K-12. Sessions will be held on topics like 'Improving Mathematics Skills,' 'Games
and Activities in Elementary School Mathematics,' 'Computers in Secondary School
Mathematics,' 'Using M in icalculators in Mathematics,' • 'Teaching Mathematics Using Multiple Texts,' 'Grade 11 and 12 Revision Update,' and many many more.
Idea Room:
The Idea Room will feature a display of mathematical ideas generated or created by
teachers and/or students in grades K-i 2. If you have techniques, student projects, activities, materials or equipment which have proven to be rewarding or motivational, please
share these with other teachers throughout the province. Wall or table displays will be
acceptable. Please send materials, or write for further information to Bob Campbell,
898 Williams Road, Richmond, B.C. V7A 1G6.
Registration:
Pre-registration is at the bargain rate of $18 until June 30, 1976. Students may preregister for $10 by the same date. All on-site registrations will be $20. For BCTF members the registration fee includes a one-year membership in the B.C. Association of
Mathematics and a subscription to Vector. Registrants who are not BCTF members may
obtain BCAMT membership for an additional $5 rather than the normal $12. Make your
cheque payable to Fifth Mathematics Summer Workshop and mail to Mrs. Linda Shortreid,
4651 - 202nd Street, Langley, B.C. V3A 5J2.
85
Northwest
Conference
by TREVOR CALKINS
Plans for the 15th Northwest Mathematics Conference to be held in Victoria are proceeding rapidly. Almost all session speakers and workshop leaders are booked. Over
100 sessions and workshops are scheduled at this time. A publicity poster encouraging
early registration should be in the mail to all schools before May 1. The final program
will go to the printer sometime in April.
Naturally, the Planning Committee feels there are many reasons why you, as a BCAMT
member, should plan now to attend the Conference. To encourage people to register
early we are offering a 25% discount in the fee. You may obtain the discount by registering before June 30 on a group registration form. This means that at least six people
must register on the same form - they do not have to be from the same school.
In the Victoria area we are encouraging schools to use Friday, October 29 as a discretionary day. Other locations on the Island and in the Lower Mainland might also consider
this possibility. Why not bring it up at your next staff meeting? If the decision is made
now it will allow members of your staff not interested in attending the Conference to
plan something else. Of course this means you would not need a substitute. Most importantly, it will allow you to register for $15 rather than $20.
The fee includes two workshop tickets and. bus transportation from the Empress Hotel
to James Bay Elementary School (5 blocks). Early registrants (before June 30) will be
able to preregister at the workshop of their choice. The first two workshops are free
for everyone - after that they cost $1 per workshop. Early registrants will also have the
choice of attending Dr. Eric MacPherson's evening session free. The subsequent Cocktail
Hour is free for those who purchase a ticket to Dr. MacPherson's speech. The price of
the cocktails is being subsidized by the Publishers Association. On-site registrants will
have to pay $1 for this privilege. Most importantly, the fee will entitle you to hear what
we feel are some of the best mathematics educators in the world today.
To guarantee your attendance we offer a list of speakers already confirmed
86
SESSIONS
ELEMENTARY
Lola May (Illinois)
Charles Allen (California)
Doyal Nelson (Ablerta)
Dora Whittaker (England)
Evan Maletsky (New Jersey)
Jim Prekesges (Washington)
Eric MacPherson (Manitoba)
SECONDARY
Father Bezuska (Boston)
E. Glenadine Gibb (Texas)
J. Egsgard (Ontario)
J. Pedersen (California)
B. Kennedy (Victoria)
M. Poage (Michigan)
H. Jacobs (California)
J. Fair (California)
WORKSHOPS
ELEMENTARY
J . Cleugh (Sooke)
Dr. K. Murray (U BC)
A. Graaf (Victoria)
C. Carruthers (Vancouver)
J . Vance (UVic)
G. Dilley (Surrey)
R. Melendez-Duke (Victoria)
E. Richmond (Queen Charlotte Islands)
I. Burbank (UVic)
T. Howitz (UBC)
W. Liedtke (UVic)
C. & D. Mullin (Oregon)
A. Coulthard (UBC)
D. Brow (Addison-Wesley)
P. George (Vancouver)
J . Sherrill (UBC)
R. Gardner (Coquitlam)
B. John (Victoria)
SECONDARY
A. Taylor (Coquitlam)
C. Long (Washington)
I. Leask (North Vancouver)
J. Nakamoto (Vancouver)
M. Parker (Victoria)
I. DeGroot (Vancouver)
C. Fleenor (California)
A. Hoffer (Oregon)
R. Makway (Alberta)
S. McGraw (Oregon)
PSA76-46
A. Neufeld (Alberta)
D. Forbes (Notre Dame)
W. Seaton (Okanagan Central)
L. Whiles (Vancouver)
L. Allaire (Victoria)
L. Raine (Victoria)
L. Shortreid (Surrey)
W. Szetala (UBC)
T. Heppell (Victoria)
I. Paul (Victoria)
J. Aston (Penticton)
T. Casey (Victoria)
G. Smith (Victoria)
C. Bannister (California)
W. Freidlander (Washington)
D. Kamp (Spokane)
S. Melnyck (Ontario)
J. Overhalser (Oregon)
B. Erickson (Washington)
S. Dunn (Washington)
A. Hoffer (Oregon)
R. Robittaile (UBC)
R. Monnard (California)
D. Seymour (California)
K. Billings (Oregon)
0. Schaaf
87