3)ov) at led A;

J,,ef

BRITISH COLUMBIA ASSOCIATION OF MATHEMATICS TEACHERS

NEWSLETTER

VOLUME 11, NUMBER 3 FEBRUARY 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, Delta. 943-1701 (home) 274-1264 (school)

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

Treasurer J. Michael Baker, 87 - 14909 - 107A Avenue, Surrey. 581 3898 (home) 588-3458 (school)

CANT 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 membership in the BCANT. Journals may be purchased at a single copy rate of $1.50. Please direct enquiries to the Publications Chairman.

BRITISH COLUMBIA ASSOCIATION OF MATHEMATICS TEACHERS OF THE BRITISH COLUMBIA TEACHERS' FEDERATION

BCAMT -- AGM

The annual general meeting of the B. C. Association of Mathematics Teachers will be held in the BCTF auditorium on Monday, March 30, from 9:00 a.m. to 4:00 p.m. The main part of the program will be a workshop on 'Probability and Statistics in General Mathematics.'

The emphasis in the workshop will be on laboratory situations in the teaching of Probability and Statistics, which should have application throughout most grade levels. Probability and Statistics are being introduced increasingly at all grade levels from Grade 3 to 12. In addition, the laboratory approach to mathematics appears to be cycling into prominence.

Our program should be of value to you in one of three ways:

1) something for general math OR 2) introduction to Probability and Statistics in the classroom OR 3) laboratory lessons and techniques

Dr. Thomas Howitz and Dr. Gail Spitler (both UBC's Faculty of Education) will develop and present the sessions. Each participant will receive a kit of mate-rial containing background information, labs suitable for class use and pro-grammed learning materials.

-.1

9:00 a.m. -- 9:30 a.m. -- General Business Meeting 10:00 a.m. -- 12:00 noon -- Probability and Statistics background

information (lecture and discussion situation)

12:00 noon -- 1:00 P.M. -- Lunch ? (see below)

1:00 P.M. -- 4:00 p.m. -- Working on lessons and labs

It would be very helpful if we could know approximately how many will attend the sessions. Participation will be free, but there may be a small -- less than $1.00 -- charge for the kit of material.

If there is enough interest, we can offer a catered lunch.

We suggest a sandwich buffet ($1.60) of turkey, ham, cheese and egg sandwiches, cole slaw and potato salad. There are cafes approximately five minutes away by car, but it may be more convenient just to stay in the building for lunch. If you want lunch, we must know in advance.

If you think you will be attending, PLEASE either return the attached form or inform the Vice-president by telephone.

1

PLEASE RETURN AS SOON AS POSSIBLE.

TO: Vice-president, BCANT, Mr. M. Wiebe, 4850 Linden Drive, DELTA, B.C.

946-6035 (Home) 594-5491 (School)

I shall attend the Probability and Statistics session.

I do/do not, want lunch at $1.60. (circle one)

NAME

ADDRESS

X70-64 mb/Feb. 17, 1970.

COMPUTER INFORMATION AVAILABLE

A new introductory book, More About Computers, has recently been made available to teachers by International Business Machines Corp. This elementary primer pre-supposes little or no technical background as it reviews the history of calculators, discusses computer operations and programming, and defines important data processing terms. Single copies of More About Computers are available free to teachers. Requests should be sent to IBM Corporate Editorial Promotions, Armonk, N. Y. 10504.

NORTHWEST MATHEMATICS CONFERENCE '70

One of the most difficult problems presented to mathematics teachers this year has been the exact location and date of the 1970 Northwest Mathematics Conference.

For the solution, invert this page.

LI - - 91 0100 D I 'VflIOIDIA

Our next issue will describe a method of solving this type of problem. Meanwhile, you're on your own - - unless you peek at the answer.

LOGIC PROBLEM

COUNTRY CLUB DANCE

The Country Club was having its annual dance. Herman, Ira, Jim and Kevin were taking their wives Alice, Betty, Candy and Doris (in no special order). From the clues on p. 2, find out who is married to whom, the last name of

1

each couple (Price, Quincy, Randall or Sanders), and the color of each lady's dress (black, blue, red or yellow).

1. Mr. and Mrs. Sanders picked up the lady who wore black, since her husband had to work late with Jim.

2. Betty and Mrs. Price selected pastel shades for their dresses.

3. The lady in blue was Ira's sister, and-they visited each other often.

4. Kevin played golf with Mr. Sanders.

5. Neither Doris nor Mrs. Randall wore the black dress, and Herman was glad his wife didn't either.

6. Alice wouldn't speak to the Sanders nor the Randalls.

LOGIC PROBLEM SOLUTION

ojA Joi opr ssip nq iOi qs put, A9 si iss sq

Os UJO JTO 4D 1StA JS1S Siq pue lejI pUe supue5 I1 o4 Nie;ads 4,usaop atis aouis 'q'r 41 s 4N i P)

siuoa i si iou 'ApueD 4OU si isis s,eij .5pue5 sel suI A3uin0 UtA M Si q Os '(fi i°) siOpues •y ou St UiA){ .lpu5u uxj si q (i nj) A DuinO e aq s -

IOU SJU5 OU Si 3"d UeULI3H si q 'o;-1q si ajItA sq f( i°) Apuej iou iou stuo(I o

puueUi 40u 91 UeU1JO llepII oq 4snLu ureu iei supue Si tueu s i .'i i°) Ilepued ou st ueu 4s-el

, stuoa ssup pu q4 aaotA suo(I '(z njD) speqs jsed

iOh aoijV pue A;q pule 'DeIq ;aaom Apue ug 311d si qs 'uo;iqj (9 P) iiepuea iou supue

i SE IOU 'ADUInO 4ou St weu 4siel s,oq urno st

IS i) IlePued iou ' ( z 1d 41 sl iou '(I np) s1pueg 4ou s uieu sej S 1 Apue ssup peq uo Apue 'io;uqJ (i nj:)) peq ui Ape . aq4 dn ppid

s.tpue5 pue '(9 np) supue o eds 4,usaop aqs uis 'ja qli a 41 UPIP ziv jeq; lez ioT si 41 njz)) Siuoj

P IP uu pue. (z n[D) Ssip pejq q; ue ,uplp Ao

DIVISIBILITY RULES

How do you tell easily and readily whether a number x is divisible by a number y? The following rules could be taught (learned, discovered) at almost any age level.

A. Early Rules A number x is divisible by:

2 if last digit even 3 if sum of digits a multiple of 3 4 if last 2 digits a multiple of 4 5 if last digit a or 5 6 if satisfies rules for 2 and 3 8 if last 3 digits a multiple of 8 9 if sum of digits a multiple of 9 10 if last digit a 0

if last n digits a multiple of

B. Composite Numbers To see if x is divisible by y where y is a compos- ite number: break y into relatively prime factors, then test for all of these factors.

e. g. , To see if x is divisible by 12: 12=3.4

Therefore, check the rules to see if x is divisible by both 3 and 4.

e. g., To see if x is divisible by 24: 24 = 8. 3, therefore check rules both 8 and 3. (Caution: although 24 = 6.4 or 12. 2, these factors are not rela-tively prime and therefore will not necessarily provide the desired results. )

C. Prime Numbers A rule to see if x is divisible by y where y is any

prime number may be generalized from the following example. Divisibility rule for 7:

is 36452 divisible by 7?

remove last digit, multiply by 2 and subtract from new num-ber

3

3645 2 4

364 1 remove last digit, multiply by 2 2 and subtract from new num-

36 2 ber 4

32 continue in this manner until the result of the subtraction is or is not clearly a multiple of 7.

The key is to find the number by which to multiply. It changes for different prime numbers: for 11, multiply by 2 for 13, multiply by 9 for 17, multiply by S for 19, multiply by 17

It is not too difficult to discover the multiplying number for any prime number. Suppose you want to find the number to use with 41: start with numbers you know are-divisible by 41. (E.g., 41, 82, 123, 410, 451) and attempt to follow the rule by trying different multiplying numbers until one gives the desired result (i. e. , an obvious multiple of 41 after the subtraction).

In this case use 451 45 1 It comes fairly readily to mul-

tiply the last digit by 4, then 41 subtract. The result is clearly

a multiple of 41, therefore the key number is probably 4. (You will have to check on a few more numbers before you bet your life, though.)

This trial and error method is not really difficult and has worked satisfactorily with a General Math 10 class.

By following these ideas, you can develop a divisibility rule for any number you wish.

4

BLINDFOLDED DIVISION

Letters have been substituted for numbers. Determine the value of each letter to solve the problem. When the letters have been arranged in order from 0 to 9, they should spell out a 10 - letter word or phrase.

SIN

IAN

ANET AIMLESS

AlL ANDES

A R I T T

AlL

R R N E S

HNE

S R K T

PAO

I A I S S

I HAS

I I S T T

I P S I

R SS

LE

BUY

MAE

BAA ASSURE

ALE ACCESS A L R C

SSOC Z E U R

S C U S Z L B Y

EACA

A R U E

L S M S L H N E

SC

E E H

MAGIC SQUARES

Complete each of the following magic squares. Each of the numbers 1 to 16 should be used exactly once in each problem. The sum of each column, row, diagonal, four corners and four center boxes is 34.

8 1 13

MOEN

Pj

2 I 15 m

14 I 7 I 9 11 41 5

1161141 3

5

Mr. Ewen wrote this article partly in response to criticism he received concerning his article Calculus for the Cantankerous in the September issue of VECTOR. He says, 'It (this article) proceeds still more fundamentally than the last one • . . ' and 'this time I have stated a position on the matter of an attitude in math teaching which I think is over-due.'

AN EXPERIENCE APPROACH TO CALCULUS

My article 'Calculus for the Cantankerous,' which appeared last fall, apparently caused some raised eyebrows in the mathematical trade. The critics' principal concern seemed to center on the necessity for the maintenance of rigor, especially in those areas of the domain that are likely to be troublesome. I hope I shall be forgiven for the observation that best results in teaching are obtained when the new lesson is based upon what the student already knows, and for the further observation that a student can be motivated more when he discusses what he can do with. an idea before he is required to concern himself with its limitations. I'm sure I could, if I took the time, provide chapter and verse to show these to be common principles of good pedagogy. I see no reason why mathematics teaching should be exempt from the application of these principles.

It is possible to teach the derivative and some of its applica-tions without concerning the student in the least with these confusing lessons (?) about zero over zero or the limit notation, and I believe the result to be a greater understand-ing of the concept itself as well as a greater ability to ap-preciate the considerations of rigor when they are introduced later. And, further, it is possible to teach the derivative when the student is armed only with the ability to make a substitution and with the concept of slope, that is, rise over run or, if you prefer, as I do, a change of y over a change of x.

Let's begin with a linear function, so that the method can be

6

assessed in terms of what we know to be true.

Consider the function defined by y 3x + 6:

When xis 5 7 9 11 13

yis 21 27 33 39 45

the change of y is 6 6 6 6

andy/xis 2 2 2 2

and nobody will be surprised. But, if we may discuss the slope of y = 3x + 6 in this manner, why not that of any function?

Consider the function defined by y = 2x2 + 3x:

when x=3 5 7 9 11. 13

y = 27 65 119 189 275 377

AY 38 54 70 86 102

AY /x 19 27 35 43 51

Now there's nothing especially sensible about these values of the slope in these intervals, until we have a good look at the interval from x = 9 to x = 11. We all find thinking in lOs easiest. 43 can be written 4(10) + 3. The last line can be written: -

= 4(4) + 3 4 (6).+ 3 4(8) + 3 4(10) + 3 4(12) + 3

In each case we have chosen an easy x from the interval in which Ly/Lx is found, and we find that we can express the slope in every interval using an expression having in each case the same structure. We may conclude that for the expression 2x + 3x, the expression 4x + 3 tells us some-thing about its slope. Try another,:

7

Consider the function defined by y = 4x 2 - 3x + 7:

when x= 3 5 7 9 11 13 y = 34 92 182 304 458 644

= 58 90 122 154 186 / AX = 29 45 61 77 93 or 8(4) -3 8(6) - 3 8(8) -3 8(10) -3

8(12) - 3

Again, we may conclude that, for the expression 4x2 - 3x + 7, the expression 8x - 3 tells us something about its slope. Students will not make very many of these in-' vestigation tables before they will tell you how to make this expression about the slope from the expression for the original function, especially if the teacher has learned how to ask the right questions.

For the functions we have discussed so far, this expression was derived from the slope of the secant. Is it generally the slope of the secant? The first step we take toward a higher-ordered function answers the question. Consider the function defined by y = x3:

when x= 3 5 7 9 11 13 y = 27 125 343 729 1331 2197 Ay = 98 218 386 602 866 Ly/Ax = 49 109 193 301 433

or nearly 3(4)2 3 (6) 2 3 ( 8 ) 2 3 ( 10 ) 2 3(12)2

For the function x 3 , the expression 3x2 tells us something about the slope. After making a few of these investigation tables, the student will tell you how to write this 'slope function' from the original function, without making such a table each time.

But that nearly says that the function we have written is not the slope of the secant. What is it then?

Without your bothering to give it a name other than 'slope function,' a few investigations can establish what you can do with it. In these investigations, the teacher must choose functions and intervals in the domain which will NOT produce

8

difficulty.

A flying instructor cannot produce a pilot until he gets his student off the ground, and it is after the student is able to control an aircraft that he has a chance in the event of difficulty. So it should be with mathematics. The student should be assisted to find out what he can do before you insert the difficulties.

The investigations should proceed as follows:

Have the student experiment to demonstrate to himself that F(b) - F(a)

S(a)< b - a < S(b). The order of the expression may change for some functions, and it is wise to choose functions and intervals of their domain where difficulties do not arise. Teach him to read it: 'The slope of the secant from x = a to x = b is always between the slope function at a and that at b. ' A few of these investigations, and the idea of the derivative, which he already knows how to find, will be loud and clear..

Let him write the expression thus: S(a)(b - a),- F(b) - F(a) S(b)(b - a), and you can have him 'knocking on the door' of the definite integral after a few in-vestigations.

Suggest to him that, when we used the word 'nearly' when dealing with a cubic function, we could have chosen an x in each interval that would make the slope function exactly equal to the slope of the secant, and you have introduced the idea of the mean value theorem. The rigor can come later and, in any event, it is meaningless unless the con-cepts are understood in a fundamental way.

I make no apology for teaching in such a way that the student can 'get off the ground without arranging for the ailerons to fall off. ' There is an order in which the segments of math-ematics can be taught, and no other order produces a result meaningful to the student, however satisfactory it may be to the teacher.

9

First, the concept must be clearly understood. I recognize the importance of the discovery method in permitting a youngster to generate new concepts. Second must come the application of the concept. He will want to know what you can do with it (what good is it? must be answered). Next can come the language of mathematics, but teaching language where no concept or usefulness exists is futility of a high order. After some language has been learned and understood, a student may be able to proceed to general-ization and logic, but not otherwise. To teach in any other order is nonsense.

Concept, Application, Language, Logic. How's that for a call?

Dr. Bullen (UBC Mathematics Faculty) wrote the following article in response to 'Calculus for the Cantankerous,' by Bruce Ewen published in the September issue of VECTOR. His reaction should become quite clear as you read his article.

(MORE) CALCULUS FOR THE CANTANKEROUS

It is a pity that Vector should have published the article with the above title. There were so many errors of fact, so many misconceptions that I feel obliged to make the following intuitive remarks by way of correction.

Unlike addition and multiplication, many operations are not defined for all numbers; division is one of these (taking the square root is another). So that even if you can write 5 o or 5±0 (or 4-6), it does not follow that this means any-thing, any more than the ability to write 'daf' makes this a word. a b is the number c, or, equivalently, the result of dividing

10

a by b is c, is defined by the equation a bc. Clearly, if b = 0, there is no c such that bc a, whatever a is. So 50 0, 0, 6+0, etc., are all meaningless collections of symbols, whereas, of course, 0= 0, 84-2 = 4, etc.

5 As a result of this, a function (or formula) expressed in terms of the above operations does not have all numbers in its domain. Thus 'v'x has a domain the set of all non-negative numbers, 1 has a domain all non-zero numbers, etc. x

When, as in many cases, the numbers not in the domain of function form an isolated set, it is of interest to describe the behavior of this function near its bad points. Thus 1

(domain all non-zero numbers) is clearly large and positive when x is near to zero and the nearer x is to zero, the larger the value of 1. This we write as lim 1 = c. On

the other hand, 1 is not large and positive if x is near to X

zero and negative; so here we say that limldoes not exist. X40 i

Another such example, but more complicated, is sin (-).

The graphs of these functions give a simple illustration of these remarks.

Now to the question at hand: the slope of the tangent to

3x3-23

y = x at x = 2, say. The slope of a chord = x - 2 this function is not defined if x = 2 - - as we would expect - - since for a chord we need two distinct points, i. e., x 2. Question: how does this (chord) function behave near its bad point? The answer is easy, since

3 - 8 = x2 x + 2x + 4 whenever the chord function is defined, x-2 i. e., if x 4 2. (In fact, the graph of the chord function and

that of x2 + 2x + 4 are identical except that the former has no point with first co-ordinate x = 2.) But the function

11

x2 + 2x + 4 is defined (and continuous) at x = 2, having the value 12. Hence urn x 2 - 8 = 12. This number is defined

X42 x - 2 to be the slope of the tangent at x = 2.

It is not always that easy: the limit may not exist - - for instance, take y I x I and x = 0; the limit may be hard to find - - take y = sin x and x = 0 when we have to find out how the chord function sin x behaves near its only bad point,

x = 0. A simple piece of geometry shows that if O<x çJ

cox x <sin_x<l, from which we get, of course, the x

lim sin x= 1. x+o x

Far from being unchanged since the discovery of these ideas, the notation involved here is a development of the last century and a half. The ideas involved are deep and complex but can be made reasonably intuitive; but this is not helped by semi-mystical invocations of the word 'infinity' and attempts to give meanings to all symbols it is possible to write down.

ANY REACTION?

The Secondary Professional Committee (Department of )ducation) met on December 17. The following was reported in Pro. D. Informational Bulletin Vol. 1, No. 5.

Some discussion of secondary school mathematics took place. The present program is critized by students and teachers for its irrelevance to life, e. g. ,

1) few university programs in maths are based on the content of senior secondary mathematics.

2) the senior secondary mathematics is too abstract for most students as it is oriented to calculus. There should perhaps he abstract mathematics taught in secondary school for those students who are oriented that way, and

12

who will continue on in their lives on a mathematic and science path. There should also be mathematics for life for the majority of students. This kind of mathematics should help individuals to purchase sensibly, to fill out income tax forms and so on. It was suggested that a distinction must be made between education which prepares for life and that which prepares students for further specific education. The secondary school maths and science pro-grams prepare students for the. latter. The relevancy of science and math to society should be the concern of 'education. l This kind of education is much more difficult to achieve than it is to teach pure or abstract maths and sciences. The present senior secondary science courses prepare students to study more science. Consideration should be given to those aspects of science which relate to life itself, i. e. , What is it that modern day science says to a citizen?

MORE SYMMETRY

from Alameda County Mathematics Educators, Spring 1969.

SYMMETRY FROM A TO Z

Lora Dalton Fairview School Hayward Unified School District

For a change of pace in the primary grades it is fun to examine the upper case letters of the manuscript alphabet with regard to lines of symmetry. Fold a piece of 8 1/2 " x 10" newsprint lengthwise and ask the children what letters can be cut from this folded paper. Let them predict as you cut the letter A. Discuss the results. Do the same thing with the letter C and newsprint folded crosswise. Discus-sion will develop as you cut any one of these letters: H, I, 0, X from a piece of newsprint folded crosswise and length-wise. (I use H, because it comes first in alphabetical sequence.)

13

Display the letters you have cut and decide on an arbitrary method of indicating which, if any, lines of symmetry are applicable. The following is a suggestion:

Letters symmetrical with respect to a vertical line will have a star.

Letters symmetrical with respect to a horizontal line will be underlined.

Letters symmetrical with respect to both lines will have both a line and a star.

The children will soon tell you how to mark each letter.

A B CD,E FG HIJKL

M N o P Q R S T U V W

Y Z

This work may be divided into as many different lessons as you like. Letters symmetrical with respect to a point may be discussed or not, depending on the interest of the class. Children may say the letter B is symmetrical with respect to a horizontal line. I accept this with a reminder of the slight variation in the State-adopted alphabet. Decide for yourself how you want to handle this exception. Decide also how to handle the letter Q. in case some child should notice that it is symmetrical with respect to a diagonal line. I believe this is not precisely true of the Q in the State alpha-bet.

14

LICENSE PLATE SYMMETRY

With a new style of license plate being introduced this year, this idea may appeal to you.

Symmetry is finding the internal lines of reflection of a figure. The use of graphing has its place, as does its applications to art as well as drawing from the masters in shape, color and form. Perhaps the following idea for the middle grades might give an idea of another way of getting at the subject - - and with variations to other grades.

An introduction to symmetry in the middle grades that I have found to be effective is use of state automobile license plates.

If one sets up with a class that the numerals and letters are the ones to be studied and recorded, a common ground for discussion and comparison is established for all. From this, work toward a successful lesson and added learnings in the drawing of these symbols. Don't be fooled - - the 0 and zero are not the same shape.

Side questions occur:

1. What letters are not used? 2. What numbers are not used? 3. How many of these shapes have two lines of symmetry? 4. Are there any with more than two lines of symmetry? 5. Which figures are asymmetric, and what is that?

Reprinted from ACME, Alameda County Mathematics Educators, Spring 1969, Vol. 2, No. 2

15

SCRAMBLED MATHEMATICAL WORDS

Each of the following words can be unscrambled to produce another word that is mathematical in nature. If your students can solve this list in about five minutes, perhaps they would like to devise their own list and send it in to us.

1. cleric 11. squeal 21. claimed 2. girth 12. tow 22. anger 3. fete 13. cape 23. chin 4. car 14. pest 24. grin 5. tear 15. mite 25. dray 6. lien 16. lime 26. signed 7. latitude 17. once 27. race 8. dad 18. not 28. sue 9. net 19. mare 29. angel

10. neo 20. smile 30. raged

WHERE DOES GEOMETRY BELONG?

Does the following (by G. L. Henderson in the Fall 1969 issue of the Wisconsin Teacher of Mathematics) apply to our situation in British Columbia?

A Case for Algebra_Algebra-Geometry

In the Spring 1969 issue of Wisconsin Teacher of Mathematics I outlined a suggested 9 - 12 mathematics program which contained one 'track, ' 'Algebra I, Algebra II, Geometry, and Pre-calculus Mathematics,' as a four-year sequence for some students. Changing the sequence from Algebra I, Geometry, Algebra II to Algebra I, Algebra II, Geometry is a departure from tradition and there seem to be sound reasons for such a change.

Colleges and universities more and more are demanding that entering students come with credit for three years of

16

high school mathematics. As a result, more and more high school pupils are signing up for a third year of mathematics. The traditional sequence, Algebra-Geometry-Algebra, was designed to insure that students taking two years of math-ematics received instruction in geometry in order to meet minimum college entrance requirements in mathematics.

Suggesting a change in sequence to Algebra-Algebra-Geometry assumes that all students programmed into the sequence will complete at least three years of high school mathematics. Other students should be programmed into different sequences.

Based on the assumption that students programmed into the sequence will complete at least three years of high school mathematics, and that other students will be programmed into different sequences according to their needs and abilities, the following reasons can be cited for adopting the Algebra-Algebra-Geometry sequence:

1. Study of geometry can be more fruitful and successful for students who have completed two years of algebra. This contention will be supported by almost all geometry teachers without question.

2. Less time is needed for 'review' in Algebra II if first year algebra has been a 'recent' experience; and Algbra II topics can be covered more comprehensively. (Since much algebra is included in pre-calculus mathematics, the course that comes after geometry, there need be no concern about students' potential lack of algebra experience after the sophomore year.)

3. The sequence Algebra-Algebra-Geometry results in higher total student achievement than from Algebra-Geometry-Algebra. This was documented in a study done by Mr. Clifford Korpi and reported in his master's seminar paper presented to the Graduate Faculty at Wisconsin State University, Platteville. Korpi's report contains details of a study comparing achievement of groups of students participating in the two sequences of courses. Tests used to measure achievement were: seven chemistry tests, the

17

Preliminary Scholastic Aptitude Test, the American College Test, and the Science Research Associates Test. Only the math scores of the intelligence tests were used. Additional details may be obtained from Mr. Korpi, mathematics instructor at Hurley High School.

4. Some mathematics teachers teaching sophomores feel more qualified to teach Algebra II than to teach Geometry. (This becomes a factor only if there exists a lack of flexi-bility in staff teaching assignments.)

CROSS SUM PROBLEM

The number refers to the TOTAL of the numbers which you are to fill into the empty squares. No zeros are used here and no number, larger than nine. Also, a number cannot' appear more than once in any particular number combina-tion. (Such problems as these are not too hard to make up - - if you do make your own, send us a copy for others.)

DOWN 1. twenty-nine 2.5. eleven 48. fourteen 2. eleven 27. twelve 49. twenty-seven 3. twenty 28. eight 52. seventeen 4. nine ... 29. twenty 53. ten 5. eleven . 30. six 54. twenty-six 6. ten 31. twenty-five 56. ten 7. eight 33. ten 57. seventeen 8. eight 34. twenty-four 59. thirteen 9. ten 36. twenty-three 61. six

10. twenty-one 40. seven 63.. fourteen 16. eleven 42. twelve 18. twenty-two 43. twelve 19. twenty-one 44. ten 22. six 46. ten 23. eighteen 47. thirteen

18

ACROSS 1. eighteen 25. thirteen 50. eleven 4. ten 26. thirty-two 51. eleven 6. seven 29. thirteen 53. thirteen 8. thirteen 31. eleven 55. twenty-five

11. nine 32. ten 57. nineteen 12. six 35. ten 58. eleven 13. four 36. six 60. seven 14. eighteen 37. twelve 61. thirteen 15. twenty-five 38. fourteen 62. seven 17. sixteen 39. seven 64. seven 19. ten 41. twenty-one 65. thirteen 20. thirteen 43. twenty-one 66. six 21. eleven 45. thirty-three 67. thirteen 23. nine 47. fifteen 68. twenty 24. eight 48. seven

[MENFAAFEEV'JOINEW.Emm MEMPRIMErAAMEEAPENE

EMPS 0000VEJENFAA VA

19

THEOREM OF PYTHAGORAS

Teachers of Math. 10 and their students may be interested in the following 'programmed' proofs of the Pythagorean Theorem.

Proof 1A a E b B

a

F

b

H

a

D b G a C

ABCD is a square each of whose sides has been divided into two segments of lengths a and b. The points of division are E, F, G, and H. Construct EF, FG, GH, and HE. Are AEH, BFE, CGF, and DHG congruent? Why? What can you conclude about EF, FG, GH,. and HE? What kind of figure is EFGH? Let the length of each side of EFGH be c. What is the area of each of the corner triangles? What is the total area of the four triangles? What is the area of EFGH? What is the total area of the square ABCD found by adding the areas of the four triangles and the figure EFGH?

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pynnf 2

B

Can you think of another way to find the area of ABCD? What is the area calculated in this way? You now have the area of ABCD expressed in two different ways. What is the logical conclusion about these two quantities? Equate the two quantities and state the final conclusion you have reached. C

c

Given: Right triangle ABC with right angle at C. Sides of the triangle are of lengths a, b, and c.

Using AB as a base, construct a square whose sides are equal toAB. Label the square ABDE. - From E construct EFJ AC. - From D, construct J!T. Extend to intersect DG at H. Are ABC, BDH,DEG and EAF congruent? Why?. What is the total area of these four triangles? What kind of figure is !JGHCF? What is the length of GH? What is the area of GHCF? What is the total area of the triangles andLIJGHCF? By adding these together we get the area of____________ Since AB = c, the area of ABDE = You have found the area of ABDE by two different methods. What can you conclude about the two quantities which express the area of ABDE? When you equate the two quantities, what is the logical conclusion?

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INO

PROCEED CAREFULLY - HEART UNDER CONSTRUCTION!

A HEART FOR VALENTINE'S DAY

1. Draw, in the center of your paper, a segment, AB, of measure 1 1/2"

- 2. Construct a perpendicular bisector, XY, of AB intersect-ing AB at point 0.

3. Measure OY 2 1/211.

4. With A and B as centers and a radius of 7/ construct arcs of circles intersecting OX at point C and AB at points M and N, respectively.

- 5. Construct a perpendicular bisector of MY and let it intersect CN at D.

C' 6. Using DY as a radius and D as center, construct MY.

7. Constrt a perpendicular bisector of NY and let it intersect CM at V.

,-' 8. Using VY as a radius and V as center, contruct NY.

9. Erase construction marks and write an appropriate caption for your geometrically constructed Valentine!

V

The Mathematics 'leacner February 1959.

Y

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VALENTINE POEM

You disintegrate my differential, You dislocate my focus. My pulse goes up like an exponential Whenever you cross my locus. Without you sets are null and void So, won't you be my cardioid?

- - Katherine O'Brien, Derring High School Portland, Maine

TEACHER'S GLOSSARY OF NEW MATHEMATICAL TERMS

(From the Bulletin of the California Mathematics Council)

SET: What you do in a chair. SUBSET: What you do under a chair. PROPER SUBSET: Sitting straight under a chair. EMPTY SUBSET: Somebody is absent. CLOSED SET: Kindergarten teachers. ELEMENT: Large animal with a trunk. CLOSURE: Last day of school. SYMBOL: Part of a brass band. BINARY: Two-headed canary. RATIONAL NUMBER: Four-day week. UNIVERSE: Poems you know IRRATIONAL NUMBER: Parent with a complaint. FRACTION: Broken bones. PLANE: Not fancy

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IT'S A PUZZLEMENT!

1. Jay solved a quadratic of the usual form (ax2 + bx + c = 0) by formula and got 2 as the greater root. Kay, with the perversity common to all students, switched the b and c when solving, and got 3 as the greater root. What was the original equation?

2. Hexagon ABCDEF is inscribed in a circle with radius 6". Now AB = BC = DE = EF = 6", and CD = 4". Find the lengths of side AF and of diagonal BE.

3. Admission to an exhibition is one dollar. Every one of the people who attended paid in coins of the realm (cents, nickels, dimes, quarters, and half-dollars) but, oddly enough, no two paid in the identical manner. How many people attended the exhibition?

ENGINEERS HAVE MORE FUN! A college professor was explaining to his class how the college was able to separate the mathematics students from the engineering students so that they would take the proper courses to prepare for their life's work. To do this, the beautiful girls on campus were lined up across one side of the gymnasium. The prospective mathematicians and engineers were lined up on the opposite side. The men were then told to advance half-way across the gymnasium and stop, then advance half of the remainder of the distance and stop, etc. , until they reached the other side, and when they did they could kiss the girls. Of course, the mathemati-cians realized immediately that this would be impossible and left. The engineers, knowing that they could get close enough for practical purposes, proceeded across the gymna sium.

- - Idaho Mathematics Newsletter

X70 - 31

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