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BRITISH COLUMBIA ASSOCIATION OF MATHEMATICS TEACHERS

NEWSLETTER

VOLUME 11, NUMBER 5 JULY 1970

BCAMT EXECUTIVE

President Leonard J. Gamble, R. R. 2, Clearwater. 674-3590 (home) 674-3220 (school)

Past President Curriculum Representative Peter Minichiello, 2484 Eddington Drive, Vancouver 8. 226-7369 (home) 736-0344 (school)

Corresponding Secretary Isobel C. Leask, 306 - 157 East 21 Street, North Vancouver. 987-9415 (home) 987-3381 (school)

Journal Editor Dr. E. MacPherson, 4474 Portland Street, Burnaby 1. 433-6551 (home) 228-2141 (university)

Publications John W. Turnbull, 5093 - 7B Avenue, Del ta. 943-1701 (home) 274-1264 (school)

Vice-president M. M. Wiebe, 4850 Linden Drive, Del ta. 946-6035 (home) 594-5491 (school)

Treasurer J. Michael Baker, 87 - 14909 - 109A Avenue, Surrey. 581-3898 (home) 588-3458 (school)

CAMT Representative Roy Craven, 2060 Willow Street, Abbotsford. 853-1888 (home) 859-2187 (school)

The B. C. Association of Mathematics Teachers publishes Vector (newsletter) and Teaching Mathematics (journal). Membership in the association is $4.00 a year. Any person interested in mathematics education in British Columbia is eligible for member-ship in the BCAMT. Journals may be purchased at a single copy rate of $1.50. Please direct enquiries to the Publications Chairman.

EDITORIAL

Here is the last issue of VECTOR for the 1969-70 year of BCAMT. I hope you find several areas that interest you. This newsletter is late, but perhaps the summer is a good time to send you a newsletter -- you will have some extra time to take a good solid look at the contents. You will also receive a journal this summer -- just as soon as it can be printed.

Last, and perhaps most important, I hope receiving a newsletter and a journal at this time will remind you that it is worth your $4.00 to belong to the BCAMT.

Remember, November 15 is the deadline for membership -- join again -- get others to join -- it is worth $4. 00.

J. W. Turnbull

-1_• Il PROBLEMS

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1. Counting the Days

Heard in a classroom after Field Day -- 'It's Thursday today,' said Paul. Wayne shook.hs head, 'It was Thursday yesterday.' 'You're crazy,' said Bob. 'Yesterday was Wednesday.' 'What about that,' said George. 'Wednesday was the day before yester-day.' Mark had the last word. 'It's Wednesday all day tomorrow. Only one was right. What day was it then?

2. How Much Left for the Holiday?

Joyce: 'Well, gee whizz. This is crazy. I figure it's only half of what you say it should be.' Dennis: (After one squint at the cash book.) 'I know who's crazy. You're 17 short in the dollars and 17 too many cents.' That explained it. What do you think was the correct total?

3. Try This on your Dolciani

16 - 36 = 25 - 45 (-20 = -20) Add 81/4 to each side:

4j zJ-/(x-i7

V

L -L Y+17)i

204' X 4

(O

16-36+81/4=25-45+81/4 Factor: (4 - 9/2)2 = (5 - 9/2)2 Square root: 4-9/2=5-9/2 Add 9/2: 4=5

Let your Grade 9s find the fallacy.

4. For Your Grade 12s

The sides of a triangle are consecutive numbers, such as 7, 8, 9. It's a unique triangle because its area is exactly twice its per-imeter (comparing square inches with inches). So what were the dimensions?

SUPPLEMENTARY MATERIALS

Geoff Homer has accepted an appointment as Co-ordinator of Sup-plementary Publications. BCAMT is going to follow the lead of the Science Teachers' Association with the publication of review or en-richment units. At present we are working on the problems of publi-cation and honoraria to authors. If you have a unit on some topic (factoring, probability, fractions, decimals, etc.) that has been useful to you, why not contact Geoff Homer at 33675 Marshall Road, Abbotsford? You may have an opportunity to help other teachers and to receive an honorarium at the same time.

PROBLEMS

1. Do you like ALPHAMATICS? Here's one to test you.

MINI

M I S S Don't forget thatM I N I would

I N certainly be ODD -- from what I've

M I N I seen. Skirt too. SKI R T

2. I have heard that this formula can be used to find the volume of any regular solid, even a sphere:

V = B + 4M + Tx H B is area of the bottom, M is the area

6 mid-way up and I is the area of the top; H is the height of the solid.

2

Try it on the sphere, the cylinder, the prism or even a frustum of a cone.

3. Here's one to test your Modern Mathematics.

Janet and Debbie were two teenagers whose birthdays fell on the same day, but their ages were not the same. Their father was a mathematician. When a visiting friend at the birthday party asked the father how old they were, the father replied, 'When Janet is as old as Debbie will be when Janet is as old as Debbie will be when Janet is as old as Debbie is now, Debbie will be a year younger than Janet will be when Debbie is twice as old as Janet is now.' Can you calculate their ages?

4. Try this one on your Grade 9s. What numeral is to be written in the last space?

IT'S A PUZZLEMENT!

• 1. If you write the first n integers in ascending order and find that the last 15 numbers are composite, what is the least value of n for which this is possible?

2. Removing 100 yards of ribbon from a (circular) spool of ribbon reduces its diameter by one inch. Removing the next 100 yards reduces the diameter further by two inches. What is the thickness of the ribbon?

3. Which has greater value: e ' or -K

MATH RECREATION PROBLEMS

1. If in ABC, a4 +0 +c4 2c 2 a 2 +b2), find C.

2. Show how to draw a chord of given length in a given circle

3

through a point which may or may not be in the given circle.

3. Solve the system:

(ax) 109 a + (by) lOg b

blog x = c log y

4. Solve the determinant --

8x+1

óx+2 9x+3 12x=0

8x+1 12x 16x+2

In making coffee in a percolator, I'd like to be able to reheat it and get one cup of coffee each of three times (breakfast, lunch, dinner). Now I know when I pour, the last cupful reaches the point marked 2" on the inside of the percolator. I also know that there is a 2% loss of fluid every time I heat up the contents of the percolator because of evaporation. To which mark mustl fill- my percolator to be sure of precisely three cups of coffee?

Equilateral L ABC is placed between two parallel lines 3" apart in such a way that A lies on one line, B lies on the other and Vertex C is 1" from one of the lines. Find a side of the L. The squre of a 2-digit number is a 4-digit number. The sum of the digits of the 2-digit number is one half the sum of the digits of the 4-digit number. The 2-digit number is twice the sum of the digits of the 4-digit number. Find the 2-digit number.

A quad inscribed in a circle has side AB=7, CD=6, BC=4, AD=9. Diagonals AC, BD meet at P. Line PH perpendicular to AB is extended to meet side CD at K. How long are segments PH and PK?

A circle with radius 10" is tangent to two sides of an equilateral L. and passes through the trisection points of the third side. Find a side of the

two are wrong.

Then two wrongs make a right. (0O)

4

With me arbitrating, çherasone.

Jn

r I

TRENDS IN THE TEACHING OF ELEMENTARY GEOMETRY

Ray W. Cleveland SMTS Journqj , Volume 7, Number 3,

University of Calgary June 1968.

I. The Approach

a) The geometry of the elementary school should be the physical geometry of the three dimensional space in which the child lives. It should make use of the observations that , he makes of the world

about him. The approach should be multi-sensory and appeal especially to the senses of sight and touch. It should be above all student activity centered and make use of pictures, cut-outs, models, paper-folding, blocks, measuring devices, and a multi-tude of other types of material, much of which the student can construct himself.

b) The spirit should be that of an investigation by way of activities that are purposeful and meaningful to the child. The activities should also be of interest and provide a degree of insight into mathematical ideas appropriate for the maturation of the child.

c) The ultimate goal should be to acquire the capacity to gen-erate new ideas from old ideas by the process of reflection.

d) Reasoning at this stage should be based on intuition rather than a knowledge of a highly structured system. Procedures to stimulate reasoning should be inductive rather than deductive.

e) One of the aspects of geometry that makes it a very suitable subject for the elementary school is the aesthetic quality to which children readily respond.

• Evidence of this is contained in the use of geometry in archi-

tecture, the use of projective geometry for perspective in drawing,

the use of 'cubism' by certain artists, and a multitude of other such examples. The composition of geometric designs and such activities as curve stitching, constructing 'space spiders,' mobiles and other objects bring out the aesthetic quality.

f) Above all at the elementary level geometry should be fun. It should be somewhat recreational and include games, stories either from the history of mathematics or designed to illustrate a con-cept, projects, contests, decorations, building models from blue-prints, and many other motivational activities. The only limit on the types of lessons on geometry(besides the time limit) is the ingenuity of the teacher.

II. The Teacher

a) The teacher should have a background in geometry substantially beyond the scope of the elementary school program. (This may seem like a superfluous statement, but unfortunately there are many teachers who have had little or no background in geometry.)

b) The teacher should be a humanist and continually aware of the response and reaction of the students. She must be sensitive to the intellectual pulse of the students. Above all she should have a sense of humor that is in good taste and appropriate to the age level of her students.

Ill. The Content

a) Topological aspects of geometry seem (on the basis of certain research) to be better understood on the intuitive level by students than many of the more restricted concepts of Euclidean geometry as traditionally taught in the schools. Topological topics suitable for elementary school are closed and open curves, exterior and interior of closed curves, betweenness, separation, connectivity, traversability of networks, and others.

b) Some topics usually quite well understood by most elementary teachers are related to geometric figures and geometric relations. Among these are such concepts as parallelism, perpendicularity, intersections and unions of geometric sets, points, lines, segments, polygons, circles, polyhedra, spheres, cones, cylinders, con-gruence, similarity, scale drawings and maps.

c) Metric aspects, such as length, area, volume and standard units of measure, are useful and instructive.

d) Co-ordinates should be introduced and used for many purposes. Single co-ordinate (real numbers) on a line, pairs of co-ordinates in a plane, triples of co-ordinates in space should be introduced by use of interesting games (such as those developed by David Paige).

e) The idea of vectors in terms of direction and distance on a line and in a plane has many uses.

TOPOLOGY

SMTS Journal, Volume 9, Number 2, March 1970.

Student: What have you in the library on Topology? Librarian: We've quite a bit. On what country did you want infor-

mation?

The following is an essary on Topology composed by a Grade 10 student, Robert Brown, of Walter Murray Collegiate, for his teacher, Doug Gleave. Your editor thought it worthy of publication even though the essay was given a B rating. Robert titled it 'The Lighter Side of Topology.'

Topology: The branch of mathematics which deals with those properties of spaces which are unaffected by changes in size and shape. It makes use of surfaces, knots, networks and many other figures.

II. A. Closed, Open and Connected Sets: If A is a subset of space, for example the set of points inside a cube, then P is a limit point of A if every neighborhood of P contains a point of A. (A neighborhood is a sphere with the given point P as center.) This means that P is a subset of A. Quoting from Collier's Encyclopaedia: 'Thus the boundary points of the cube are the limit points of the interior points. However, if P is distinct from a point Q, then P is not a limit point of the set consisting of Q alone.'

A set is closed if it contains all the limit points, hence a line is a closed set. The set of all points not lying on a given point is an open set.

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If a set is decomposed into two sets with at least one point in each set and if one set contains a limit point of the other, then the sets are connected.

B. Topographical Changes: In topology a doughnut is more related to a coffee mug than to a pretzel. A coffee mug and a doughnut are said to be 'homeomorphic' or 'topologically equal.'

A figure can be transformed into an equal figure in topology. This is because the genus number of a doughnut is equal to that of a coffee mug while different to that of a pretzel'.

The genus number of a topological figure may be found by two mfhrdc!

1. The holes in the figure may be counted, the number of holes being the genus number.

2. You may count the number of cuts which may be made without dividing the object or figure.

Sphere Torus Joined Rings No holes - Genus 0 One hole - Genus L Two holes - Genus 2

No cuts - Genus 0 One cut - Genus 1 Two cuts - Genus 2

(Note - Objects do not fall apart with cuts as indicated.)

Therefore, a cube is equal to a sphere, a torus is equal to a coffee mug, and two joined rings are equal to a sugar bowl.

8

(From Life Science Library - Mathematics. Also, Bergamini, David: Mathematics - New York, Time Incorporated 1963, page 178.)

III. A. Topological Models: 1) One sided Surfaces: - Most people are familiar with the Mobius Strip. It was introduced by a German Mathematician, Augustus Ferdinand Mobius in 1858. It is simply a strip of paper which has been given a half twist and the ends fastened together. This produces a curious one-sided, one-edged loop. If cut lengthwise you don't get two loops but a single long one with two half twists. If you cut it into thirds, you get two intertwined loops, a long one with two half twists and a small Mobius Strip. If you draw a line completely around the loop, you would find that the line would be continuous and on both sides of the loop.

The double Mobius Strip is somewhat different. It is constructed by placing one strip on top of another. This is given a half twist and the ends are fastened together. It appears as though you have two loops, one seated inside the other. What you actually have is a continuous loop identical to the one produced when a regular Mobius Strip is cut in half.

The Klein Bottle is also along this line. It was invented by Felix Klein in 1882. This also has only one side, -an outside but no inside. It is the only figure that can be cut in half to produce a Mobius Strip.

uble twist wist

Edge is a simple closed curve.

Formation of the Klein Bottle (From the Scientific American).

The properties of the Mobius Strip and the Klein Bottle have been summarized in the following two limericks. (From the same article in the Scientific American.)

A mathematician confided That aMobius Strip is one sided And you'll get quite a laugh If you cut it in half, For it stays in one piece when divided.

A mathematician named Klein Thought the Mobius Band was divine.

Said he, 'If you glue The edges of two

You get a crazy bottle like mine.'

2. Some Interesting Topological Models.

Diagrams 1 and 2 are one sided and one edged. Diagrams 3 and 4 are two sided and one edged. Diagrams 5, 6,. 7, 8, 9, and 10 are one sided and two edged. Diagrams 11 to 16 are two sided and two edged.

Edge is a simple closed curve. Edge is knotted.

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Edge is knotted. Both edges are simple closed curves, un-linked.

Both edges are simple closed curves, linked.

7 8 9

10

Both edges are

Both edges are

One edge is simple, knotted, unlinked. knotted, linked. one knotted, linked.

One edge simple, Both edges simple

Both edges simple one knotted, linked. closed curves, closed curves, linked.

unlinked.

12

Both edges are knotted, unlinked.

13

Dth edges e knotted, nked.

15

One edge is simple, one knotted, un-linked.

16

One edge is simple, one knotted, linked.

II

B. Flexagons: These are interesting shapes made by folding and gluing strips of paper. They have the property of changing faces when flexed. Here are shown diagrams for the construction of a simple trihexaflexagon.

/ \ / \ / \ \ \ Z /,\2. /3\3/_\Z//\//'3"\ \1 \\// \\,/ \j/

A.

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As the name trihexaflexagon indicates, it has three faces. To change faces, pinch two adjacent triangles together and push the opposite vertex in toward the center. The flexagon should now open up, exposing a new face.

The flexagon was discovered by Arthur Stone in 1939. Because he had to trim American paper to fit his English binder he had some-strips of paper left over. He started folding the scraps in various ways. One shape he found would change faces.

His flexagon had three faces. After some thought he decided that one could be made with six faces rather than three. He confirmed this and told some of his friends about it. They formed a club to study flexagons and found that by lengthening the chain of triangles, the number of faces could be increased by multiples of three. One was made with forty-eight faces. It was also found that flexagons having four, five, seven and eight faces could be made by using crooked sheets of paper.

C. (Editor's Note: At this point Robert showed how an inner tube of outstanding stretching capacity could be turned inside out if the valve stem were removed. He made a number of diagrams from Life Science Library, Mathematics, which makes this item too long for publication. He, however, added this interesting foot-note. 'Note that any line which formerly traveled the length of the tube now travels about the circumference of the tube.')

D. Networks. To a topologist a knot is uninteresting. Networks, however, have a natural topological interest.

The oldest of these - the Koenigsberg Bridge problem is shown below. The citizens of Koenigsberg liked to stroll on the bridges

The Koenigsberg Bridges

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and the question was proposed: How could one walk across all seven of the bridges without crossing any one twice? They soon found that it was impossible but didn't know why.

In 1736 Leonard Euler solved the problem of 'why' by putting out the network theory. The principle behind the theory is: 'the number of journeys necessary to travel the network is equal to half the number of odd vertices, (that is, the vertices with an odd number of segments ending at it).' It is not possible to have an odd number of odd vertices.

E. The Four Color Problem: Although it seems simple enough that anybody could solve it, nobody has, or proved that it is in-soluble.

On a map, each country is colored a different color from that of any other country that is touching its borders. The minimum number of colors must be used.

The problem is to prove that four colors are sufficient on a plane or a sphere and that it is impossible to draw a map which needs five colors. But if a map is placed on a Mobius Strip it may require six colors. If the map is rolled into a torus, it may require seven colors. V

Conclusion

Topology is interesting and fun, and provides a relaxing (?) pastime or hobby.

Bergamini, David: Mathematics -- New York: Time Incorporated, 1963.

Collier Encyclopaedia, Toronto: P. F. Collier and Son, 1959.

Gardener, Martin; The Scientific American Book of Mathematical Puzzles and Diversions, New York, Simon and Shuster, 1959.

Tucker, A. W. and Bailey, H. S.: Topology, New York: Scientific American Incorporated, January 1950. V

15

COMMON DENOMINATIONS MADE EASY

Illinois Council of Teachers of Mathematics, Volume 20, Number 4.

By Jerome T. Filipek, York Community High School, Elmhurst

The amount of effort which is spent on teaching the addition of fractions and the idea of common denominators probably exceeds that spent on any other topic in the study of fractions. The problem of finding the lowest common denominator plagues many students through high school. Those working with high school freshmen and sophomores still find too many students who cannot add fractions, other than those of the simplest forms. It is evident that these students have not acquired any method for finding the LCD (lowest common denominator). Obviously the lack of student skill in doing this type of problem can-not be blamed entirely on substandard teachers or on student inat-tention. There must be something wrong with the methods now employed in the teaching of common denominators. The purpose of this article is to introduce a method which had been found successful for most of the students to which it has been presented.

Not long ago I was involved in a discussion with my students on the adding of two or more fractions. The discussion centered around the problem of rapidly finding the LCD for the numbers 32 and 108. I started off on the idea of introducing the LCM (lowest common mul-tiple) and proceeded to divide out all common factors from 32 and 108.

32 108

16 54 (dividing by 2)

8 27 (dividing again by 2)

After dividing by all of the common factors I had a pair of numbers which truthfully, at first, I wasn't too sure had anything to do with the problem. I knew that 8 and 27 were relatively prime and should have something to do with the LCD. It quickly became apparent that since 8 = 7, then 8 x 108 32 x 27 = 864, and that since

32 108 8 and 27 are relatively prime, this product of 864 must be the LCD. From the example we found that the LCD of any pair of numbers could easily and quickly be calculated using this method.

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Assume we have the problem 5/24 + 7/60. Have the student con-struct a list of the denominators of the two fractions.

24 60

Then find any number which evenly divides 24 and 60. (Note that if such a number does not exist then the two denominators are relatively prime and the LCD is their product.) List the quotients below their respective denominators.

24 60

12 30 (dividing by 2)

The students may choose different divisors. This may cause them to have different tables, but the final quotients will be the same. Brighter students will naturally see larger, more complex divisors than slower students. The list might proceed as follows.

24 60

12 30

2 5 (dividing by 6)

The final table quickly leads the student to the LCD by finding the product of the last entry in the first column and the first entry in the second. column (or the first entry in the first column and the last entry in the second column). In this case 24 x 5 2 x 60 120 and we have found the LCD. This method allows the' student a quick check of his work as well because the two products must be equal.

The same basic method is applied in wórkng with more than two fractions. A lowest common denominator is found for a pair of the fractions. Using that LCD you repeat the process with another de-nominator. In this way you can safely calculate the LCD for any set of fractions.

Students have been quick to grasp this technique and use it with understanding and effectiveness. The idea of making tables can be used to explore other ideas as well. Students who find abstractions difficult have success doing mathematics this way.

Mr. Filipek is a student of Mr. Charles Schulz at Northeastern Illinois State College.

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DECIMAL FORMS OF RATIONAL NUMBERS

SMTS Journal, Volume 8, Number 1, December 1968.

Fr. Albert H. Ruetz, St. Peter's Junior College, Muenster, Sask.

When treating the decimal forms of rational numbers, a few added details often help to make this section more interesting and not just mechanical manipulations. Most of these details might just be men-tioned in passing, and not overstressed or once more the tedious mechanical manipulations will far outweigh any interest they might have engendered.

In Book 1 of Modern Algebra, Dolciani mentions: 'A decimal with a finite number of place ... is called terminating, ending, or finite,' (page 401), e.g., = .25, 1/5= .2, 1/64= .01562, 21/16= 1.3125. Do these fractions have anything in common which makes them ter-minating? For a fraction to terminate when expressed as a decimal, it is necessary that the denominator consist only of powers of 2, 5, or both. Each of the above examples has this characteristic. Numbers such as 2, 4, 5, 8, 10, 16 1 20, 25, 32, 40, 50, 64, 80, 100 -- to mention all those not exceeding 100 -- consist only of powers of 2 and 5, and therefore if these numbers occur on the denominator, the decimal form of these numbers will terminate. A brief statement and application of this fact can make the statement in Dolciani come to life. A teacher could get members of the class to give fractions with denominators less than 100 and then state whether they terminate or not, and in this way draw this fact to the attention of the class.

'The decimal form of any rational number r/s either terminates or eventually repeats in a block of fewer than sdigits.' (Dolciani, Book I, p. 401; Book II, p. 185.) Question 29, Book I, page 403 is an interesting question to 'play' with. 1/7-2/7 ... 6/7 can be assigned to different .students in class with the following results.

1/7= .142 857 4/7= .571 428

2/7= .285 714 5/7= .714 285

3/7= .428 571 6/7= .857 142

After you have received the decimal notation for 1/7, the decimal notation for all the others can be written down. You will notice that a cyclic arrangement exists and that each succeeding fraction begins wi th the next larger number to the one preceding it. 1/7 begins with

18

i f 2/7 with 2, 3/7 with 4, and so on with the same cyclic arrangement occurring in all. The corresponding digits in both 'halves' always add up to 9. Only 9 of the 25 primes below 100 -- 7, 17, 19, 23, 29, 47, 59, 61, 97 -- have a period of one less than the number itself when its reciprocal is expressed in decimal notation. 1/97 must be carried to 96 places of decimals before the cycle repeats. All of these have a decimal expansion which is cyclic and whose 'halves' add up to 9. The basis for this lies in whether 10 is a primitive root of the number. The reciprocal of any prime having an even number of digits in its period has this same property. 13, for example, has a six-digit period, and has two sets of cyclic arrangements.

1/13 = .076923 2/13 = .153846 3/13 = .230769 5/13 = .384615 4/13= .307692 6/13= .461 538 9/13 = .692307 7/13 = .538461

10/13 = .769230 8/13 = .615384 12/13 = .923076 11/13 = .846 153

Primes that have an odd number of digits in their respective periods do not have the above properly. Again, there are 9 of these less than 100 --31, 37, 41, 43, 53, 67, 71, 79, and 83. The recip-rocal of a composite number which has an even number of digits in its period may or may not have the digits in the half periods com-plementary.

Again, too many of these details cannot be added in a class, but to show this properly for a/7 should add something to the interest of the class. 'The decimal form of any rational number either terminates or repeats.' Students may be challenged (not as homework) to show that such is actually the case with 1/97, 1/61, or 1/59. Few will have the patience and many will make mistakes before they get a period of 96, or 60, or 58 respectively. It is suprising how some will persevere. In any case it will make students appreciate the work of W. H. H. Hudson who carried out the computations of 1/1 861 to its full period of 1860 digits and of William Shanks who did the same for 1/1 7389 to its period of 17,388 digits.

Reference: Beiler, Albert H. Recreations in the Theory of Numbers -- The Queen of Mathematics Entertains, Dover Publications, Inc., New York, 1964.

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BC 1 AD AC = CD DE E any point on arc

SCIENCE PUZZLERS, Scholastic Book Services -- March 1967, is another.delightful magazine full of 'Fun Ideas,' as well as instructional tidbits that students would enjoy reading.

One idea is for an angle trisector. This is one of the three classic problems of ancient Greek mathematicians that cannot be solved using Euclidean tools, straight edge and compass; but students can have fun with this little gadget.

Given: Angle XYZ less than 90 degrees. Figure is not accurate.

Place: Point A on initial side of angle, with side BC passing through vertex Y. Point E will be point of tangency of terminal side XY.

If Angle to be trisected is greater than 90 degrees, bisect the given angle and proceed as above.

20

BRAIN TEASER

Here is an interesting one that will test your logical abilily. All the facts needed to answer the questions posed are contained in statements 1-15 below.

1. There are five houses, each of a different color and inhabited by men of different nationalities,, with different pets, drinks and cigarettes..

2. The Englishrnan'lives in the red house.

3.' The Spaniard owns the dog.

4. Coffee is drunk in the green 'house.

5. The Ukrainian drinks tea.

6. The green house is immediately to the right of the ivory house.

7. The Old Gold smoker owns snails.

8. Kools are smoked in the yellow house.'

9. Milk is drunk in the middle house.

10. The Norwegian lives in the house next to the man with the fox.

11. The man who smokes Chesterfields lives in the house next to the man with the fox.

12. Kools are smoked in the house next to the house where the horse is kept.

13. The Lucky Strike smoker drinks orange juice.

14. The Japanese smokes Parliaments.

15. The Norwegian lives next to the blue house.

Now, who drinks water? Who owns the zebra?

(from Mathematics Flyer. - Minnesota)

21

ARE ORDERED PAIRS OUT OF ORDER?

The Bulletin of the California Mathematics Council, Volume 27, Number 1, Fall 1969.

By James R. Smart, San Jose State College

The use of ordered pairs in developing new number systems is becoming common at almost every grade level. For rational numbers, the number pair (3 1 4) represents 3/4 and (a,b) represents a/b. The existing con-vention for naming rational numbers as ordered pairs is that the first number in the ordered pair represents the first number in the indicated division and the second one indicates the second number, so that (3/4) represents 3 divided by 4. The purpose of this article is to explore the consequences of reversing the order in the ordered pairs for rep-resenting rational numbers.

Suppose that the rational number 3/4 is named by the ordered pair (4,3) rather than (3,4). What is to be gained? The first advantage is that this arrangement corresponds much more closely to the intui -tive development of fractions in elementary grades. In old fashioned language, the denominator is used to show the size of each part, while the numerator indicates the number of those parts. Thus, (4,3) implies breaking something into four ports, then naming three of those parts. The numerator depends on the denominator, rather than the other way around. Also, when fractions are used for measurements, the denominator indicates the unit of precision, so that naming it first is convenient.

A second and even more significant effect of reversing the order results from the fact that an ordered pair used to represent a rational number meets the definition of a function. A function is a set of ordered pairs (x,y) such that no two pairs have the same first element. If the ordered pair (4, 3) represents the rational number 3/4, then the graph of the function (4,3) is the point such that the slope of the segment from the origin to the point is 3/4, as in Figure 1. Similarly, the point for 5/7 is the point (7,5) on a segment with slope 5,7 and the point for a/b is (b,a) on a segment with slope a/b. Note that all the points naming the some rational number (such as (4, 3), (8,6), etc., naming (3/4)) are on the line through the original with that rational number as slope. Reversing the order in ordered pairs would help to connect the concepts of rational number, function, and slope

22

X

y

of a line. An added advantage is that points on the y axis, whose slope is undefined, would be associated with number pairs such as (0,3) resulting from division by zero, also undefined.

F 19U1

A third advantage of reversing the ordered pairs for rational numbers is that one can tell which of two rational numbers is greater simply by graphing the ordered pair and comparing slopes. In Figure 2, you can easily tell that the segment from the origin to (5,4) has a greater slope than the segment from the origin to (3,2), hence 4/5 is greater than 2/3. The rational number is greater if the slope is greater, so that the answer can be found simply by looking at the relative positions with respect to the origin. This idea, it should be pointed out, is currently used in British experimental programs at the junior high school level. i

Mgure 2

Though mathematicians may not reverse the order of ordered pairs for rational numbers, mathematics educators nevertheless can enjoy thinking about the consequences and can learn more about number systems by exploring results other than the three included here.

23

x

HOW THE PRIMES GOT DOUBLE-CROSSED

The Bulletin of the California Mathematics Council, Volume 27, Number 1.

by V. C. Harris, San Diego . State College

Although it took about 2200 years, the primes got double-crossed. Here is the story.

The positive integers can be divided into three parts: (1) the unit 1, and (2) the primes 2, 3, 5, 7 ... whose only positive divisors are themselves and 1, and (3) the composites 4, 6, 8, 9, 10 ... which have positive divisors other than themselves and 1. The determination that a number is a prime, or is not d prime, is an interesting and dif-ficult problem1.

Eratosthenes is the first actor of our drama. He lived about 250 B. C. was'-famous as an athlete' geographer, historian, astronomer and mathematician. He invented an ingenious device for duplicating the cube2 , but his main achievement for purposes of ourstudy is his 'sieve, '3 whose description follows.

Write in order the odd positive integers, beginning with 3, as for as desired: 3579111315171921232527293133

Moving (one place) to the right in the array corresponds to adding 2 to the number. Adding 2 or 4 to a multiple of 3 gives a number not a multiple of 3, but adding 6 to a multiple of 3 gives a number which is a multiple of 3. Hence every integer of the array which is a multiple of 3 (except 3 itself) is crossed off if every 3rd number after 3 is crossed off. 3 is a prime. The next number after 3 not crossed off is 5, a prime. Moving to. the right corresponds to adding 2 to the number. Adding 2, 4, 6, or 8 to a multiple of 5 gives a number not a multiple of 5, but adding 10 = 5 . 2 to a multiple of 5 gives a number which is a multiple of 5. Hence every integer of the array which is a multiple of 5 (except 5 itself) is crossed off if every 5th number after 5 is crossed off. Hence crossing off every 3rd number after 3, every 5th number after 5, every 7th number after 7, etc. leaves only primes not crossed off. Thus the primes (except 2) less than 33 are given in the array:

24

35791113 15 17 19 21 23 25 27 29 31 33 and this can be con-tinued as far as desired.

After a lapse of centuries, Valerio (1893) noticed that only the prime ending in 5 is 5 itself. His scheme, which follows, gives all the primes except 2 and 5. Write the odd numbers prime to 5 in four rows according to the endings 1, 3, 7, 9 (as far as desired):

1 11 21 31 41 51 61 71 81 91 101 111 3 13 23 33 43 53 63 73 83 93 103 113 7 17 27 37 47 57 67 77 87 97 107 117 9 19 29 39 49 59 69 79 89 99 109 119

Moving to the right in the array corresponds to adding 10 to the number. Adding 10 or 20 to a multiple of 3 gives a number not a multiple of 3, but adding to a multiple of 3 gives a number which is a multiple of 3. Hence every integer of the array which is a multiple of 3 (except 3 itself) is crossed off if in every row the first multiple of 3 (except 3 itself in the 2nd row) and every 3rd number after it is crossed off. The first number larger than 3 not crossed off is 7, a prime. Adding 10, 20, 30, 40, 50 or 60 to a multiple of 7 gives a multiple of 7, but adding 70 10 . 7 to a multiple of 7 gives a number which is a multiple of 7. Hence every integer of the array which is a multiple of 7 (except 7 itself) is crossed off if in every row the first multiple of 7 (except 7 itself) and every 7th number after it is crossed off. The continuation with 11 -... leaves only primes not crossed off, except for the unit 1:

111 21 31 41 51 61 71 81 91 101 111 3 13 23 33 43 53 63 73 83 93 103 113 7 17 27 37 47 57 67 77 87 97 107 117 9 19 29 39 49 59 69 79 89 99 109 119

Since this scheme requires fewer crossings off than the previous method, Valerlo may be looked upon as a villain in our drama.

In 1908-09 Morehead extended the sieve to numbers of the form m ak + b (m 1, 2, 3 ...) and as before, from our present point of view, this improvement was a step backward.

Instead of crossing off, we will underline, for purpose of clarity. Instead of underlining, we could write numbers as in this scheme: Start with (say)

25

12345678910111213 Write 1 under each number. Write 2 under 2 and under every second number after 2. Write 3 under 3 and under every third number after 3. Continue for 4 1 5 ... as far as the array goes. We would have 1 2 3 4 5 6 7 8 9 10 11 12 13

2 3 2 3 2 7 2 3 2 11 2 13

4 3 495 3

6 8 10 4

6 12

From this we could find the sum of the divisors of any number . n in the array, usually denoted by o(n), and the number of positive integers

- n, and relatively prime to n, usually denoted by o(n). (But we shall not do this.)

Write down the integers in order: 123456789101112131415161718 Cross off 1. and every integer after 1. Cross off 2 and every second integer after 2. Cross off 3 and every third integer after 3. Continue as far as the array extends: 12 3 4.5678 9 10 11 12 13 14 15 16 17 18

Then each integer is crossed off a number of times equal to the number of its divisors (usually denoted by (n)):

(a) Squares and only squares are crossed off an odd number of times.

(b) Composites and only composites are crossed off 3 or more times.

(c) The unit and only the unit, 1, is crossed off exactly once.

(d) The primes and only the primes are crossed off exactly twice.

References 1. V. C. Harris: 'A test for primality.' To appear in Nordis

Matematick Tidskrift. 2. For a different method of duplicating the cube, see: E. V. Graef

and V. C. Harris: 'On the solutions of three ancient problems.' Math Magazine, 42 (Jan. 1969)pp. 28-32.

3. For references to Eratosthenes, Valerlo and Morehead, see

26

L. E. Dickson: History of the Theory of Numbers. Volume 1, pp. 347-356.

4. V. C. Harris: 'A Modification of the Sieve of Eratosthenes.' American Math Monthly, LX (May 1953), pp. 325-6.

STATEMENT OF BCAMT AS OF APRIL 30, 1970

Balance, June 30, 1969 (overexpended) $237.31-

Receipts: Membership Fees $2,132.00 Membership Fees (late) .96.00 Membership -- BCTF Grant 11312.00 B. 0. X. -- NCTM Grant 248.94 B.O.X. --BCTF Grant 250.00 In-service -- BCTF Grant 8th N. W. Conf. 99.50 Publications and Advertising 25.15 Workshop -- Sales 92.25 Workshop -- BCTF Grant 37.50

Total Receipts 4,293.34

Disbursements: Executive --

-meetings (7) $532.95 -postage 19.80 -stationery 76.27 -telephone 7.40 -constitution (600) 16. 80 653.22

9th N.W. Conference -advance $200.00 -planning mtg. 31.50 -postage . 5.50 237.00

Workshop (March 30/70) 301.43

8th N. W. Conference -delegate $199.00 -advertising 43.56 242.56

27

Publications -newsletters (4@660) $552.88

-journals 348.98 -J. Bouman 13.29 915.15

CAMT Conference (R. Craven) 300.00

B.O.X. (allotted $498.94) 234.81

Sundry -NCTM dues $ 6.72 -fee refund 4.00 10.72

BALANCE ON HAND $11163.14

BCAMT BUDGET PROPOSAL

July 1, 1970 - June 30, 1971

Receipts: Membership 550 @ $4.00 $2,200.00 BCTF Grant 250@ 3.00 750.00

250@ 2.00 500.00

• $3,450.00

Expenses: Journals 3 @ $350.00 $11050.00 Newsletters 6 @ $130.00 780.00 Math Contest 100.00 Exec. Mtgs. 8 @ $100.00 800.00 Delegate to CAMTor In-service 200.00 Spring Workshop 100.00 Delegates to 9th N. W. Conf. 100.00 Secretarial Expense 50.00

$3,180.00 •

Contingency: 270.00

$3,450.00

28

REPORTED IN PRO. D. (BCTF) Volume 1, Number 6.

Arithmetic/Mathematics

The Revision Committee is aware that there has been an apparent drop in computational skills in elementary school students because of the introduction of 'modern mathematics.' In connection with this, the comment was made that modern mathematics in the elementary school teaches a language, which is not subsequently used in secondary mathematics. Secondary mathematics deals with 'real' numbers. The drop in computational skills, if it does indeed exist, may be due to a misinterpretation by teachers of the nature of the present curriculum, popularly known as 'modern mathematics.'

The committee is also aware of the need to consider the introduction of the metric system into the elementary mathematics program. This question will be carefully looked at.

INTERESTING DISSECTION PROBLEMS

by Dr. Howard Eves

(Reprinted from ACME, Alameda County Mathematics Educators, Volume 1, Number 2.

'It is interesting,' said Dr. Eves, 'that in some fields of mathematics one must study for years and years before finding himself on the "frontier" but in geometry, one can be on the frontier in a few minutes.' To illustrate his thesis he presented a few interesting theorems and results -- and unsolved problems -- from the field of dissection theory.

For example, an early conjecture held that it is impossible to dis-sect a square into smaller squares, no two of which are congruent. Although this conjecture has been proven false, no one has been able to accomplish the dissection using any fewer than 24 smaller squares. If you would like to become famous, simply find a more economical solution, or prove that such does not exist.

29

Figure 1

B

Figure 2

A B

A general theorem states that an equilateral triangle can be dis-sected into 5 pieces which can be rearranged to form a square. How-ever, Doodly ( --- 1931) discovered the 4-piece solution given in Figures 1 and 2: C

Figure 1: Given an equilateral triangle, ABC, of side s, bisect sides AC and CB in points E and D, respectively. Strike an arc from point D, with radius equal to the length of the side of a square whose area is the some as that of the original triangle (SJ15, intersecting side AB in F. 2

C

I-Figure 2: Locate G so that FG CD (length s). Erect perpendiculars EK and

2 GL. If the triangle is cut along these lines, the resulting 4 pieces can be arranged into the shape of a square. After you have tried to thus rearrange the four pieces, (and perhaps given up), try this:

30

establish a 'hinge' at each of the points E, D, and G.

Figure 3 A

Now keep quadrilateral AFKE immobile. (See Figure 3) Grasp point F of triangle FGL and swing it counterclockwise clear around the quadrilateral back to its starting position. The triangle is now a square.

Fill in the missing digits so that is 'checks out' horizontally and vertically. Only the digits 1-9 are used.

-- .--

--. _

_• _

SOMEONE.Is there more than one solution? See: A Collection of Cross Number Puzzles, J. Weston Walch, Publisher, Portland, Maine.

31

Can you group these numbers in such a way that the combinations of 2, 3, 4, 5 and 6 numbers, respectively, each results in a total of 100. Each number should be used just once and none should be left at the end. Use only addition.

Fill in the missing operations (+, -, , x) so that it 'checks out' horizontally and vertically.

• _

• ____

ii i•

Is there more than one solution?

32

9th NORTHWEST MATH CONFERENCE

October 16 - 17 1 1970 Victoria, B. C.

The conference will be held at the Oak Bay Junior Secondary School in Victoria. At present, confirmed speakers are:

Miss Lola May, Winnetka Schools Irving Adler, N. Bennington, Vermont Father S. J. Bezuska, Boston College Mathematics Institute A. P. Shulte, Oakland Project Ray W. Cleveland, University of Calgary Eric MacPherson, University of British Columbia S. A. Jennings, University of Victoria

Plan to attend this conference. You will be well rewarded in many different ways.

THE CANADIAN ASSOCIATION OF MATHEMATICS TEACHERS

Roy Craven is the BCAMT representative to the Canadian Association of Mathematics Teachers. You are a member of CAMT through your membership in BCAMT. Here is some information about the CAMT.

In 1960 1 the Canadian Teachers' Federation sponsored an Invitational Mathematics Seminar, which brought together delegates from all of Canada. One of the recommendations of this seminar was that a Canadian Association of Mathematics Teachers should be formed. Little was done at that time to follow up the recommendation.

In March 1967, the CTF convened a conference of mathematics teachers on the topic Mathematics and the Teacher. As part of this conference, the feasibility of forming a Canadian Association of Mathematics Teachers was again examined. Guidelines for the formation of such a group had been passed by the CTF's AGM in 1966. In accordance with the guidelines established by the CTF, the con-ference formally created the CAMT.

The Planning Committee of CAMT, elected at the March Conference,

33

met in June to draft a constitution and to finalize plans for a second conference scheduled for December 1967. The topic chosen for this conference was Mathematics in Canadian Schools. At the conference each province was given the opportunity to describe the development of its mathematics curriculum. Also, during this conference, the draft constitution for CAMT was ratified and an executive council was elected.

The Executive Council convened a meeting of all the CAMT councel-lors in March 1969. This meeting began planning for the next con-ference. This conference, on the topic ETV and Mathematics, was held in Toronto in April. At this conference each province had the opportunity to describe what is being done there withETV in the mathematics classroom, and to explain the present state of ETV development and what trends are foreseen.

The main purpose of the Canadian Association of Mathematics teachers is to provide a means of communication between the teachers of mathematics in the provinces and territories and to co-ordinate some common activities. It has often been. hinted that Canadian education is too greatly influenced by educational systems in other countries. Many Canadian teachers of mathematics know more about curriculum trends in other countries than they know about those in Canada.

Many common problems are being faced by mathematics teachers across Canada. For example, as the curriculum demands continue to change, how does a practising classroom teacher 're-tool' or keep up? What about the evaluation of pupil progress at any level, especially as the trend continues toward more and more individual-ization of instruction? What is being evaluated? What about content -- some mathematics is necessary for all, but not the same kind of mathematics for all students -- who selects? from what? where? And the questions flow endlessly!

We know what is being done to solve some of these problems --nothing! But in some areas, some interesting attacks are being made -- for example, on the question of in-service training for the practising teacher. In some provinces the Department of Education has a team of experts which travels around the province. When the in-service team arrives in a particular area the schools are closed for a day or two at a time, and the teachers attend intensive in-service workshops. In other provinces, such a touring in-service team is sponsored by the

34

Teachers' Federation. The workshops are held in the evenings or on weekends and attendance is voluntary. One province, however, has as part of its in-service team a number of substitute teachers who take over while the regular teacher attends the in-service workshop. In other provinces, other approaches to 're-tooling' are being tried out. We should know about them and about how successful they seem to be. Telling us is one of the jobs that CAMT is attempting to do.

What is being done with teacher aids, mathematics laboratories, sup-plementary materials, computer assisted instruction, ETV, etc.? Once again, the list seems endless and the role for CAMT to play increases in importance.

The teachers in British Columbia have a lot to offer to the teachers in the rest of Canada and, by the same token, they have a lot to gain from the others. CAMT can help and is attempting to do so by providing the means of communication among the teachers of math-ematics across Canada.

Since the formation of CAMT, there has been an increasing acceptance of this group by other mathematical groups in Canada. For example, the Canadian Mathematical Congress has been working very closely with CAMT, to the point where many of the CMC sub-committees have CAMT representation on them. One such joint effort resulted in the Canadian Mathematical Olympiad held in May 1969. (It should be pointed out that the CMC is basically a university-industry mathematics group. One of its sub-committees is concerned with school mathe-matics, and it is this group that has welcomed the formation of CAMT because a closer liaison between mathematics personnel at all times will now be possible.

In August 1969 the First International Congress on Mathematical Education was held in Lyons, France. The Canadian delegates to this Congress included two representatives from CAMT. A report from these persons was presented at the CAMT General Meeting in Toronto on April 19, 1970.

35

A REPORT BY THE ARITHMETIC/MATHEMATICS REVISION COMMITTEE

as reported in Volume 1, Number 10 of Pro. D. (BCTF)

The Arithmetic/Mathematics Revision Committee has beenaskéd to plan a revised mathematics program for elementary and secondary schools. for recommendation to the Department of Education. In its work to date, 'the committee has-examined the strengths and weak-nesses of the present program; considered representations from post-secondary institutions, the business community, and interested teachers; and examined the programs of other jurisdictions. In addition, a number of other factors and their possible effect on the mathematics program have been discussed. These include:

-the nature and purpose of curriculum guides; -the background and training of teachers; -the likely consequences of new organizational systems, including semester systems;

-the possible impact of computers; -the nature and purposes to be served by elective mathematics courses; -the metric system of measurement.

Early in its deliberations, the committee reached agreement in prin-ciple that there is both a body of mathematical knowledge and skills that should be provided for certain groups of students. As a con-sequence, the committee has based its planning on a possible two-part mathematics program. The first part would extend to at least the first year of the secondary school and would be essentially the same in content for all pupils. The second part, except for one or two corn-pulsory courses, would consist of elective courses selected by the pupil because-of:

1. post-secondary goals; 2. secondary school program, requirements; 3. pupil interest and ability. -

The precise point of division between the two parts, to be in terms of content rather than age or grade level, is still under consideration.

A detailed scope and sequence for the first part of the program is

36

presently being planned by the committee. This is being based on the following nine strands:

1. Number and numeration 2. Geometry 3. Measurement 4. Applications of mathematics 5. Statistics and probability 6. Sets 7. Functions and graphs 8. Logical thinking 9.. Problem solving

Considerable time has been spent in considering the role of geometry in the total mathematics program. The committee has agreed that an integration of geometry with arithmetic and algebra is preferred. Where integration is not feasible, an interleaving of the geometric topics is preferred. The committee has agreed in principle that an intuitive presentation of Euchdean geometry should be completed by the end of the first or second year of the secondary school. The place and role of formal deductive geometry, Euclidean or otherwise, has still to be decided.

The committee wishes to emphasize that the foregoing is provided as an indication of the present state of the work that is proceeding. It must be clearly understood that no final agreements have been reached nor have firm recommendations been made to the Department of Education.

WHEN IS A GENERALIZATION?

by Bruce Ewen, Carson Graham Secondary School, North Vancouver

You may not agree with some of the preliminary observations and thoughts, but I think you will appreciate the text of the following by Bruce Ewen.

Teachers of mathematics are too often prone to proceed from definition and unexplained generalization, when better learning could result from providing the means whereby students could understand what is being done, and out of which some of them could produce the

37

generalizations for themselves.

It is probable that we do this because in most cases the texts we. use proceed in this fashion. This manner of 'teaching' is also found to be fashionable at the universities, so who can blame a workaday teacher in a secondary school for following a pattern so widely used by people obviously more 'successful' than himself? It is also fashionable to recognize a generalization only if it is couched in abstruse symbolism. While this may be acceptable at the university level, I doubt if it has any place in secondary schools.

The textbook we use in B. C. (Modern Algebra and Trigonometry, by Dolciani et al.) is probably one of the best used anywhere at the Grade 11- 12 level, yet it teaches a great many lessons as a result of unexplained generalizations. I think the worst example is its treat-ment of matrices and determinents. The entire topic of matrices is completed without once permitting the student to participate in the development of the generalization, but I suppose this is not surprising for according to Eric Temple Bell, Cayley did not explain how he arrived at his use of the matrix, nor explain how he arrived at such definitions as matrix multiplication. He was satisfied to state and then prove. When a teacher is blessed with students who ask, 'How come?,' he must be prepared to do a little digging, for there's precious little information to answer the student who wants to see the source of the ideas.

In this book the determinant is taught as a consequence of the matrix, in spite of the fact that they were used many years before Cayley's 1858 publication, notably for finding areas and volumes in Cartesian co-ordinates. I leave to you the question of whether or not it is good pedagogy to present topics historically out of order. When finally the determinant is used, in this chapter, for the solution of several equations in several unknowns, the lesson begins with a problem in third-order determinants; and the student finally participates in writing the generalization -- but through two pages of solid symbolism. I believe the following is a more understandable approach to both determinants and matrices.

Consider this ordinary algebra problem, the solution of the system:

3x + 5y = 4 7x + 9)f 6

38

Before proceeding with the solution, put another question: Aren't we really trying to transform the system so that it reads:

x + Oy = something Ox + y = something

and wouldn't it be grand if we could do it in a single step?

Now, let's prepare the equations for the elimination of one of the unknowns. But instead of performing the arithmetic operations, let's save them up until there's nothing left to do. Here they are:

To eliminate y: 9(3)x + 9(5)y = 9(4) 5(7)x + 5(9)y = 5(6)

Now subtract:

[9(3) - 5(7)]x = 9(4) - 5(6) and solve:

x= 9(4) - 5(6) 9(3) -5(7).

To eliminate x: 7(3)x + 7(5)y = 7(4) 3(7)x + = 3(6)

[7(5) _3(9)]y = 7(4) - 3(6)

y=7(4) -3(6) 7(5) -3(9)

The rest is observation. First, the denominators are opposites. Let's change y so that its denominator agrees completely with that of x.

X = 9(4) -5(6) y = 3(6) - 7(4)

9(3) -5(7)

3(9) -7(5)

The fractions that give x and y are made entirely of coefficients of the original equations. You can write

X = 4(9) - 5(6) y = 3(6) - 7(4) 35

3 5

A 79

7 9 in which the denominators ape the appearance of the coefficients in the left members of the system if only you will say that

35 79

means 3(9) - 7(5). Simply a different manner of representing -8, but using the appearance of the original equations to guide you, you could, if you wished, now write

39

45 3 4 x6 9 and y 7 6

-8 -8

noting that to find x, the x-coefficients in the array in the denomin-ator are replaced by the constant terms, and to find y, the y-coef-ficients are so replaced.

But there's .a more important observation to be made. The questions at A could, if you chose, be written

X = 9(4) + (-5)(6)

y = (-7)(4) + 3(6) 35

35 7 9. 79

that is, as the sum of two products rather than the difference of two products, and if you will say that number-patterns can be 'multiplied' this way:

1

9 -5 1 [4 1 - ( 4)9 + (-5) (6) 7 3] [oj - L(-)() + 3(6)

you will have produced the numerator of both x and y by 'multiplying' the right members of the system by a pattern not unrelated to that of the left members. The numerators of the solution sef are thus available at a single step.

I believe that in this way teachers can improve their students' under-standing of a topic and-permit some of them to find the generalizations for themselves. The idea is much. like using an experiment in a science class so that every student may find, from his own observation, that which made Newton write his second law. It is not quite the same, however, since the lab of the mathematician really is his own mind -- he deals with ideas, not physical properties.

Just one question. Is the foregoing development a generalization? I do not expect many to agree that it is, since I chose to discuss a particular system of equations. But if I had written A, B and C, in-stead of 3, 5, and 4 respectively, they would have appeared in the solution set in the same positions as 3, 5 and 4. You then would have agreed that the discussion was completely general. Not a bit more revealing, but general. I do not think the choice of symbols important, as long as it is understood that any symbols could have been chosen. More students will follow, and more will find themselves

40

able to go beyond the bare bones of the lesson and do their own generalizations.

ANOTHER LOGIC PROBLEM

The president of the Green Thumb Garden Club rapped her gavel on the desk. 'Ladies, come to order, please!' she said. 'I am very pleased to announce that at this year's District Flower Show, our group won five ribbons: two blue ones, a red one, a white one, and one Honorable Mention.' All the ladies applauded, for this was the best they had ever done. The president continued, The ladies respon-sible for the ribbons are Rose, Pansy, Iris, Violet and Daisy.' The ladies again applauded. Oddly enough, the arrangements that won were of flowers the same as the ladies' names: there was an arrange-ment of roses, one of pansies, one of irises, one of violets and one of daisies. But no lady designed an arrangement whose name was like her own. From the clues below, determine each lady's winning flower arrangement, the ribbon she won, and her last name: Austin, Barkley, Caudle, Daniels or Eden.

1. At the show the arrangements were grouped according to the variety of flowers: Group A contained all arrangements of roses; the arrangement that won the white ribbon was in Group F; Mrs. Barkley's arrangement was in Group K; Daisy's arrangement was in Group R; the arrangement of irises was in Group W.

2. Mrs. Caudle placed higher than Mrs. Eden; Mrs. Eden placed higher than the lady who arranged pansies.

3. Daisy and Violet had won blue ribbons in previous years, so both were a little disappointed that they had not won one again this year.

4. Mrs. Daniels and one of the blue ribbon winners rode together to the state capital for the show. The lady who arranged roses, Iris, and Violet followed in another car.

5. Neither Pansy, nor Mrs. Austin, nor the arrangement of violets won a blue ribbon.

X70-3/nc

41