Inequalities IAuthor(s): Eleanor DeanSource: The Mathematics Teacher, Vol. 78, No. 8 (NOVEMBER 1985), p. 590Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27964658 .

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1. Both proofs and programs theoretically begin with pictures? diagrams and flowcharts. Both dia grams and flowcharts are aids like a blueprint. Both are interpreted and expanded as the writing fol lows. Both use special mathemat ical symbols.

2. Both proofs and programs are written in numerically num bered sequences called statements and in lines from top to bottom (at least in BASIC).

3. Proofs and programs have a

preferred starting method. Proofs list part, if not all, the given facts first. Programs also begin with def initions and given values (i.e., "Let").

4. The best proof or program consists of the least number of statements or lines, with no faulty logic resulting from omissions. " Nothing extraneous, nothing

omitted."

5. While writing both a proof and a program, one must keep the last step in mind. Both require a certain amount of backward think

ing, since, in both instances, the

"prove" or end is prior knowledge ("Where do you want to end up?").

6. Both proofs and programs re

quire a prescribed "language." Proofs require theorems, axioms, terms, and symbols set within the confines of a contained geometric logic ; computer language has its various commands and symbols. It isn't possible to write a proof or

program without a prior introduc tion to the necessary wording.

7. Another less obvious simi

larity is what is done with a proof or program once completed. Com

monly used proofs are numbered, often named, and saved or set aside as a "proved theorem" usually listed with other such proofs in the back of a geometry text. Programs that are frequently used are also named. They are the programs and

subprograms that are named and saved in the computer's memory or a library.

8. You can buy a geometry text that comes equipped with all the

accepted common theorems. These

"prepared" theorems are used to build and solve other proofs and

problems in the text and for partic ular students. You can buy com

puter software that comes with all the common programs. These pre pared programs can be used to write customized programs for a

particular user.

In discussing geometric proofs, the teacher can encourage the use of computing terminology. This practice can also help students in

consciously or subconsciously "overlaying" and relating the two

logical compositions. For example, "go to" is an easy term to include when directing students to a par ticular part of a proof (i.e., "Go to statement 6"). The word line can be interchanged with statement on occasion (i.e.,

" Go to line 6").

Error is a better term than mistake or wrong. The computing terms you incorporate into your geome try vocabulary (i.e., let, next, print, if-then, home, end, for, read, save, goto) will vary according to the

language the students have been (or will be) exposed to.

Mary L. Rogers A. W. Beattie Technical

School Allison Park, PA 15101

Inequalities I In the February 1985 issue (p. 91), Sandra C. McLaurin offered "A Unified Way to Teach the Solution of Inequalities." I have had moder ate success using this approach. Teachers contemplating using the "unified" method ought to be fore

warned, however, that students

may jump to the (invalid) con clusion that once the critical

points are determined, the solution set always contains only alter

nating intervals. Of course, this conclusion is not true, as the fol

lowing example shows :

(3 - x)(x - l)2(x + 2) > 0 Its graph is

It is interesting to consider the effect of repeated factors having odd multiplicity and also the effect of using (3

? x) instead of (x

? 3).

Eleanor Dean Tallahassee Community

College Tallahassee, FL 32301-8170

Sandra C. McLaurin responds : Dean's comment that students often jump to invalid conclusions is correct. I attempted to stress in

my article that the student should test the truth of the statement on each interval. When teachers pre sent material on inequalities, they

should include as many examples as time permits, including exam

ples with repeated factors such as these :

(x - 2)2(jc - 3)(x + 4) > 0 (even multiplicity)

?h-1? -4 2 3r

and

(jc - 2)\x - 3)(x + 4) > 0 (odd multiplicity)

4 -4

Examples with factors of the form (x

? 3) and (3

? jc) should be

included to illustrate that the solu tions occur in a variety of ways. Yet, with good students it is very desirable to talk about general conclusions concerning the solu tions of inequalities when certain kinds of factors are present. Unfor

tunately, there is seldom time in most algebra classes to cover the material in the depth that is desir able.

Inequalities II Although I am an ardent exponent of unifying mathematics in any way I can, I must take issue with Sandra C. McLaurin ("A Unified

Way to Teach the Solution of In

equalities," February 1985). First, I disagree with the start

ing position of the article. Prob lems 1 and 2 are related but in a

more profound way. The following theorems are given without proof to my intermediate classes in al

gebra but are proved formally in the advanced classes.

Theorem 1. [/(*)]* < a2 ~ \ f(x) \ < a<-> ?a < f(x) <> a, a > 0

Theorem 2. [ f(x)]2 > a2 | f(x) | > a++f(x) > a or f(x) < ?a, a > 0

It is stressed that the less-than

problem leads to a conjunction, whereas the greater-than problem leads to a disjunction. Situations

where a is zero or negative (not covered in the article) are dis cussed. The relationship?the square root of a perfect square of a number (or expression) is the abso lute value of that number?is sa lient in the theorems.

(Continued on page 651)

-Mathematics Teacher

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