SEEKING FORMULAS FOR TABLES OFRELATED VALUES

LAURA BLANKHugJies High School, Cincinnati, Ohio

In the larger school systems, usually classes are graded as toI. Q. or percentile rank, or as to earlier achievement in a subjectin the same field of study. Consequently there may be as manyas four levels of ability in as many classes. The group of highestpossible achievement should be one of our most challengingeducational studies. To be honest with ourselves, we must ad-mit that we probably put our least efforts and energies there,for they will pass the course of study, with a minimum of ourguidance. Moreover the mediocre group, or J^-group so ex-hausts our plans as to subject matter and method and in-dividual need that only occasionally do we scintillate, sendsparks that reach latent powers in our highest ability group.

In a beginning course in algebra the attempt to work from atable of related values to a formula or verbal statement of con-ditions is an experience perhaps little developed. Yet reasoningof this kind is of our most valuable experiences in life as intel-ligent individuals. Thinking that makes for progress whether inthe purely scientific field, of the research type, or of a moregeneral character, is of the analysis, then synthesis type. Con-sider the work of the Mellon Institute of Pittsburgh, an or-ganization of diagnosis, its thought processes are of this type:analysis many times repeated, then attempts at synthesis.Theirs is no simple, fortuituous trial and error endeavor todetermine conclusions.

Is not much serious human thinking an effort to find patterns,patterns of things, patterns for social relations, for humanconduct? Is not the highest idealistic, spiritual form of endeavor,the effort to find the divine purpose and plan, pattern for one’sown life? Moses, in Hebrews 8:5 was told: "See that thou makeall things" according to this pattern showed to thee on themount." This life design was to be made effective and operativein human life. Our upper intellectual stratum, no doubt, con-tains many of our leaders of the immediate future. They mayremake the world.Our class is one in elementary algebra or general mathe-

matics. All effective teaching of algebra to-day is, to some de-gree, of the nature of general mathematics; Simon pure algebra

867

868SCHOOL SCIENCE AND MATHEMATICS

belongs to previous decades. Our class has progressed in ourcourse of study, or unit, if you prefer, so as to be able to trans-late a simple formula into a table of appropriate and relatednumber values. Perhaps the pupils have developed formulasfrom verbal statements, and, then from the formulas, tables.Many such relations can be profitably studied. There are thoseinvolving uniform speed of travel, uniform cost of articles,those concerning areas and volumes, and those involving aver-ages. There are many more sorts.Now, conversely, to develop formulas from tables of related

values, that is a much more difficult reasoning process, theacme of scientific endeavor.Let us consider problems graduated in difficulty:There remained in a storage tank only 18 gallons of water on

a certain day. It was then filled at the rate shown in the follow-ing table:

12Number of minutes =

Number of gallons =

0

18

1

30

2

42

3

54

4

66

5

Develop a formula for the number of gallons in the tank afterany given number of minutes, using g and m.The parcel post charges for the third zone (from 150 to 300

miles) are represented by the table:

12Number of pounds =

Number of cents =

1

9

4

15

5

17

8

23 31

Evolve a formula expressing the cost c of p pounds.The rate on telegrams to a certain town is represented by the

following table:

Number of words =

Number of cents =

10

50

12

56

15

65

20

80

Express as a formula the cost, c, of w words.From the table below, develop a formula which will tell how

y depends upon x:

x==

y=

i

2

2

0

3

-2�

4

-4

5

-6

6

-8

FORMULAS FOR RELATED VALUES 869

The following table represents the cost in cents of sending par-cel post packages into the sixth zone. Derive the formula interms of c and p:

Number of pounds =

Number of cents =

^12

2

19

3

26

4

33

5

40

6

47

7

54

8

61

Formulas may frequently be obtained from tabular values byinspection, as we probably have been doing here. If, however,values appearing in the table are complicated, and it is expectedthat the formula be of the linear type, it may be found easilyby using the equation: c=mp-\-n. In the example above19=2m-}-n, and again, 4:0=Sm-\-u. Solving these two equationsin m and n, we find m=7 and n=5. Hence c==7p-{-5. We arepresupposing a graphic understanding of the linear equation.However our formula may not be linear.Develop the formula connecting x and y:

x==

y=

0

6

1

6.8

2

7.6

3

8.4

4

9-2

During certain weeks of a vacation period, a boy^s savings indollars increased as shown in the following table:

Number of weeks =

Number of dollars =

1

31

4

^5

6i

7

n9

9iDevelop a formula in d and w,

Possibly we should close our study by presenting tables ofvalues not in linear relationship, perhaps such as to producequadratic formulas, or maybe no known formulas. Our youngstudents, perhaps a bit conceited by their success in findingthought patterns among mathematical relations, need knowsomething of humility. Yet, no one can be taught humility. Itis a purely subjective experience. Perhaps no group of our pres-ent day thinkers knows it more convincingly than our greatscientists. An example is the negro, George Carver, of TuskegeeInstitute.’ A poet once said as he plucked "a flower in the crannied wall,"and gazed at it in his hand: "If I could understand what youare ... I should know what God and man is.^