understanding absolute value < 0

Understanding Absolute Value <0Author(s): Stephen C. SinkSource: The Mathematics Teacher, Vol. 72, No. 3 (MARCH 1979), pp. 191-195Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27961585 .

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success on each trial). If time permits, I would also like to include transposition codes?codes for which the letter frequency of the code is the same as that of the normal

message. More on this and a newly in vented type of code can be found in the

August 1977 issue of Scientific American in the "Mathematical Games" column.

Now, if I can only stop my students from

passing notes in code. . . .

Solutions to Codes

Homework question 3. Coming soon: how some codes are broken, or how statis tics are used to solve a cryptic.

Table 2. One secret of breaking a code is to know which letters and words are most often used. The other is to persevere until the code is completely broken.

Table 4. Another important aid in read

ing substitution ciphers is derived from the

fact that cipher alphabets are usually based on some system or key.

REFERENCES Gardner, Martin. Codes, Ciphers, and Secret Writing.

New York: Simon & Schuster, 1972.

-. "Mathematical Games." Scientific American 236 (August 1977):120-24.

Kahn, David. The Code-Breakers. New York: Mac

millan, 1967.

Laffin, John. Codes and Ciphers?Secret Writing through the Ages. New York: Abelard-Schuman, 1964.

Sinkov, Abraham. Elementary Cryptanalysis, A Mathematical Approach. New Haven: Yale Univer

sity Press, 1975.

Wolfe, James. Secret Writing?the Craft of the Cryp tographer. New York: McGraw-Hill, 1970.

Zim, Herbert. Codes and Secret Writing. New York: William Morrow & Co., 1948.

James Feltman South Ocean Avenue M.S.

Patchogue, NY 11772

Understanding Absolute

Value The title of this article is trying to suggest

that the most haphazardly presented and least understood topic in secondary school

mathematics is the solving of open sen tences involving absolute value. This is un

fortunate, since these problems offer such a

complete review of first-year algebra tech

niques. I am going to demonstrate a method that

my students are using to solve open sen tences with any number of absolute value

expressions. It is critical that the students

first understand the definition of absolute value:

= if > 0 = -

if < 0

After discussing my method of solving equations, I will review the second part of the definition, which gives students the

most difficulty. So that we agree on vocabulary, the ex

pressions that appear inside the absolute value symbol are called the interior. For

March 1979 191



example, - 4 is the interior of | jc

? 4|.

The conditions of the interior are ? 4 > 0 and : - 4 < 0.

The most important part of this method is to realize that the solution set(s) of the conditions of the interior(s) must be inter sected with the solution set(s) of the open sentences that result when the conditions are substituted in the original open sen tence. If this is not done, then there is no

simple method to determine whether the solutions will be roots of the original open sentences without substituting for the vari able.

The solution of each open sentence must be treated as a series of cases. The number of cases depends on the number of ex

pressions involving absolute value in the

open sentence. If there is one absolute value

expression, then there will be two cases to consider. Two expressions yield four cases, and three expressions yield eight. In gen eral, if y equals the number of absolute value expressions, then there will be 2y cases to consider. Many of the cases may yield null results, but they still must be treated with equal respect.

Figures 1, 2, and 3 indicate a pattern that

may be used to solve any open sentence

involving absolute value. The cases in each

problem result in the intersection of the solution set(s) of the conditions with the solution set of the open sentence that re sults when the conditions are substituted in the original open sentence. The final solu tion set of the original open sentence is the union of all the final intersections in each case.

When teaching this topic, I encourage the exclusive use of graphs instead of the rule form so that students gain a pictorial understanding of the process. After they see how different problems are solved, they be

gin to take shortcuts. Earlier I mentioned that the second part

of the definition of absolute value causes considerable confusion. In a way this is not

surprising. In the elementary grades, stu dents are accustomed to thinking that the absolute value of a quantity is equal to a

nonnegative value. Later they are told that

I jc I = if > 0,

which they learn without questioning be cause there is no negative sign preceding the to the right of the equal sign. Most students simply ignore the condition that

> 0. Then the confusion begins, when

they are told that I jc I = ~x if < 0.

Case I Interior conditions:

Substitution:

x-2>0

{ : X > 2}

5x-{x-2)= 22 Ax = 20

\x: X = 5}

Case II Interior conditions: jc -2 <0

{jc: jc<2>

Substitution: 5jc - [- (jc -

2)] = 22

6jc = 24

{ : X = 4} -

U *

Fig. 1. 5x ? I jc ?

21 = 22 (Two Cases)

192 Mathematics Teacher



Case I

Interior conditions: - 1 > 0 and + 4 > 0

{x: X> 1} {*: X> "4) =

{ : X > 1

Substitution:

(x- l) + (x+4)<6 2x + 3 < 6

{x: x<3/2} Case II

Interior conditions:

x-l>0andx + 4<0

{x: x> 1} U- *< -4} =0 Substitution: (unnecessary following )

( -1) + ( + 4)]<6 Ox - 5 < ?

Ox < 11 { : xeR|

Case III Interior conditions:

- 1 < 0 and + 4 > 0

{ : X < 1} {*: >

{x:

-4} =

4 < < 1} Substitution:

"( -

1) + ( + 4)<6 + 5 < 6

< 1 { : R}

-

?

j

Case IV Interior conditions:

x-l<0andx + 4<0

{ : X< 1} { : < "4} =

{ : < -4} ?

Substitution:

+ [-(* + *)]< ? "2 -3<6

( : > -9/2}"

U 3

2

?> -

1

4

Fig. 2. | -

11 + | + 4| < 6 (Four Cases)

They see the negative sign preceding the

and immediately conclude that this con

tradicts their idea that absolute value is

nonnegative. As teachers, we are only fool

ing ourselves if we believe that the average student does any more than read the condi

tion jc < 0 or even understands its signifi cance. Here is where the teaching must be

gin. Few first-year algebra students under

stand that a negative sign preceding a vari

able does not necessarily imply a negative

value. The key is the replacement set for the

variable. A teacher must carefully show

with examples that if < 0, then ~jc, the

opposite of , is nonnegative. For example, if = ~4, then =

~(~4) = 4. Using

~(x -

6) as another example, ~(x -

6) is

nonnegative if jc - 6 < 0 or < 6. If we

choose some value like jc = 4 to check this

result, then we see ~(x -

6) = 2.

One of our greatest weaknesses is realiz

ing that we are seeing problems through the

eyes of many years of experience and that

March 1979 193



Case

1

Interior conditions*.

y- 1 > 0and + 3 > 0

{(x,y)>.y>i andx> "3}

Substitution:

(y-~?}<(x + 3)+? \

y<x+5 11 {(x,y): y<x+5)-J

Case

II


y- \ >0andx+3<0

\{x, y\.y>\ and < ~3}

Substitution:

l)<-(*+3)+1

- 1 .

{(*,;,): ^<-x- 1}-'

Caselli


j> - 1 < 0 and + 3 ? 0

{( , y): y< 1 andx^-3}-^

Substitution: 1

-(v-i)<(x + 3)+r

3 1

1

{{x,yy. y>~x-3\-/

Case IV


y- 1 < 0 and x + 3 < 0

{(x, y)i y<\ and x< "3}

Substitution: /""N

"0>-1) <""(*.+3)+1 <(*,j)? J>>*+3}-'



the student is seeing the problem for the first time. This is where the art of teaching begins.

Stephen C. Sink Central Davidson Senior High School

Lexington, NC 27292

Recognizing Quadratic Equations with No Real Roots

When my students studied quadratic equations and discriminants, I asked them to generate their own examples of equa tions with two distinct real roots, a double real root, and no real roots. Everything went smoothly until they tried to give ex

amples of equations that have no real roots.

Finally they did find a method that yielded an infinite number of quadratic equations with no real roots. Their method was in

triguing because it related several different

topics they had studied, namely, using the

discriminant, solving quadratic inequal ities, and applying the quadratic formula.

As a start, they discovered that the fol

lowing equations had no real roots:

Ijc2 + 2x + 3 = 0 22 - 4(1)(3) < 0

2x2 + 3x + 4 = 0 32 - 4(2)(4) < 0

~3x2 + ~2x + "I = 0

(-2)2 -4(- 3)(-l) <0

They began to wonder if every quadratic equation with consecutive increasing in

tegral coefficients had no real roots. The general equation for this pattern is

ax2 + (a + l)x + (a + 2) =

0,

where a is a nonzero integer. The discrimi nant of this equation

(a + l)2 -

4a(a + 2) has to be negative if its roots are not real. Therefore a search was made for solutions to the following inequality:

(a + l)2 -

4a(a + 2) < 0 a2 + 2a + 1 ~ 4tf2

- %a < 0 ~3a2 - 6a + 1 < 0 3a2 + 6a - 1 > 0

Using the quadratic formula to find the roots of

3a2 + 6a - 1 = 0,

they found

a = "

?} ^?

Since they were interested in the integral values of a, they approximated the roots to be 0 or ~2. Testing the intervals a < "2, "2<#<0, tf>0, they determined that the solution to the inequality must bea < "2 or a > 0.

Therefore the integers that could not be used for a were 0 (already eliminated), ~1, or '2. If a has a value of "1, then the

equation ~x2 +1=0 has roots 1 and "1. For a =

"2, the equation ~2x2 - = 0 has 0 and ~\ as roots. All other quadratics with

consecutively increasing integral coeffi cients have no real roots.

They knew that multiplying an equation with integral coefficients by any nonzero number produces another equation with the same solution set. For example, finding that

(1) lx2 + 8x + 9 = 0

has no real roots allows us to say after

multiplying (1) by i that

7 9 j-x2

+ 4x + ^

= 0

has no real roots. Multiplying (1) by we see that

7>/3 2 + 8 /3 + 9y/3

= 0

has no real roots. Thus from (1) we can

generate an infinite number of quadratic equations with irrational coefficients that have no real roots.

The students continued their search for

March 1979 195



understanding absolute value < 0

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