ScHOOLSciENCE

MATHEMATICS
VOL. XXI. No. 5 MAY, 1921 WHOLE No. 178

TEACHING FORMULAE IN THE JUNIOR HIGH SCHOOL.

BY Jos. A. NYBERG,
Hyde Park High School, Chicago.
Ability to use a formula is now regarded as one of the most
important bits of knowledge to be acquired by every pupil from
his high school mathematics. Since examples of the formulae,
which can best be used in the ninth grade are found in every
text book, the following paragraphs consider some other aspects
of the subjectchiefly how the subject may be taught.
The mathematics of the ninth grade differs from the pupiPs
previous work in two respects, (1) the use of negative numbers
as well as positive, and (2) the use of letters to stand for num-
bers. Since the latter notion is usually introduced first and as
many formulae do not involve negative numbers, it is possible
to introduce formulae the first week of the year. If we say
sugar costs ten cents a pound so that the cosfc, c, of n pounds is
lOn, then c == lOn. This may be a simple illustration of a
formula; but a poorer illustration can hardly be found. The
relation is so trivial that the pupil may well become indignant
and disgusted at the teacher for making such a mountain out
of a molehill, and the pupil can not be blamed for losing interest
at such an obvious result. The problem must command the
pupiPs respect. A more interesting and also very simple
problem is that of dividing a number into two parts having a
given ratio. This problem can be handled during the first-
week, and I present it here in some detail just as it is presented
in the class:
^A carpenter wishes to divide a board 16 ft. long into two
parts such that one part will be twice as long as the other. If
the board were 15 ft. long this would be an easy matter as he
can readily guess that one part should be 10 ft. long and the
other 5 ft. But since the board is 16 ft. long he marks off three
410 SCHOOL SCIENCE AND MATHEMATICS

equal divisions on his board. How long is each division, all

three divisions being equal? He then saws off the board at one
of the division marks. How long is each piece? If the car-
penter wants one piece to be three times as long as the other
piece, into how many divisions should he mark the board? Afc
which one of the marks does he saw the board? If the board
were 18 ft. long how long is each division? (And parenthetically
I ask some inattentive youngster just how many pencil marks
the carpenter has made on his board. This is to help the pupil
visualize the performance.) How long are the two pieces
after sawing? If he wants the ratio of the two pieces to be 4
to 3, in how many equal divisions does he mark the board?
Where does he saw it? How long are the pieces?^
Then after working a few arithmetic exercises, I ask the class
to write in complete sentences the directions for dividing a
board into two pieces having a given ratio, each pupil choosing
his own length and a definite ratio. The directions are read
aloud and criticized. Next I produce from the cupboard a board
about two or three feet long and a saw, saying, li! am not going
to tell you how long this board is, but I wish to saw it in two
pieces having a certain ratio, and I do not wish to tell you the
two numbers in the ratio. Can anyone give me the directions
for doing the work?^ Some one volunteers, and I ask the rest
of the class to write down what he tells me, that there may be
no disagreements afterward. The pupil usually says: Add
the two numbers in the ratio, and divide the length of the board
by that, etc. When the sawing is completed we discuss what
the pupil said and conclude that he should have said: (This I
write on the blackboard.)
Call the length of the given board I, and the given ratio a to
b. Find (a+6). Find l/(a+b). Find aX l/(a+6) and bX I/
(a+&). These are the lengths of the desired pieces. ;

On the previous day when evaluating some algebraic quan-

tities we had learned that aX b/c = ab/c and so we can add
that the lengths of the two pieces are al/(a+b) and bl/{a+b).
Next I point out this significant fact: when working the problem
by arithmetic we divided I by (a+b) first, and then multiplied
the result by a; now we see that if we were working a hundred
such problems in a speed test, it would be advisable to multiply
a by I first and then divide this result by (a+^); that is, the
letters .have shown us a quicker way of getting the results. Hardly
by arithmetic would it have occurred to us to multiply the
 FORMULAS IN THE JUNIOR HIGH SCHOOL 411

length of the board by one of the numbers in the ratio. Such

a problem as this commands the admiration of the pupils be-
cause the results seem both useful and unusual.
About a week later we will be solving simple equations such
as x == John’s age, 3x = Henry^s age, etc., and so I repeat some
problems like the one above, x = length of one piece of the
board, 3rc = length of other piece, 4:x = 18, etc. And as extra
work for bright people in the class I assign the problem of
showing that x = al/{a+b) can be found from the equations
in the same way that we discovered it experimentally. Also
in every geometry class, when a line is to be divided into parts
having the same ratio as two given lines, all of this work is
repeated with emphasis put on the algebraic work as well as
on fche geometric construction. The reader will notice the cor-
relation of the subjects, arithmetic, algebra, geometry; and to
the writer it seems that this is the kind of correlation most
worthy of consideration.
The next use of any formulae comes a few weeks later when
exponents have been introduced, and we are to use the relations
about areas and circumferences of circles. At this point I have
what may seem a rather undignified and untruthful way of
introducing the letter ^. I tell the class that one day a Greek
discovered that whenever he divided the circumference of a
circle by its diameter he always got the same number for a
quotient. The number became so famous that a special letter
of the alphabet was assigned to it, the Greek letter ^r. The
number became famous because it had so many uses; for ex-
ample, if the square of the radius was multiplied by this miracu-
lous number, the result would be the area. We write this as
A = Ti-r2. And if the radius is cubed and then multiplied by
4^/3 the result is the volume of the sphere, or V = 47^r3/3. At
this point we actually introduce the word formula meaning the
most concise way of stating how something can be computed. Our
first problems deal with A and V. The relation C = 2"-r is
used last and least because we happen to be studying exponents
and because this relation seems obvious considering how 7r was
introduced. By that I mean that the pupil can find the cir-
cumference without consciously thinking of any formulae In
using any formula whether dealing with a circle or the horse-
power of a gasoline engine it is advisable to devote some time
^o make the number TT really meaningful to the pupil I have each one bring to class a tin
can, and deternun-e the value of TT. by finding the circumference with the aid of a tape measure,
and the diameter by measuring the greatest vvidth of the can.
412 SCHOOL SCIENCE AND MATHEMATICS

to encouraging neat and systematic work. Thus, in finding the

volume of a circular cylinder whose height is 15 and the radius
of whose base is 12, the following presentation may be used:
144
22
288
Given: r == 12, h == 15. Find V. 288
Use: V = Trr^h, TT = 22/7 -
31680
V == 22/7 . 144 . 15 = 6788.6 15840
7 J47520
6788.6
Mental arithmetic, such as writing 144 instead of 122 should be
encouraged also. The arithmetical work should never be done
on some other paper and then thrown away but should be
shown alongside the other work. The use of the word given
and find has many advantages and will make the later work in
geometry seem less artificial. As some problems will use
^-=22/7, others 3.14, or 3.141 or 3.1416 (according to the
character and number of decimals of the given numbers) the
pupil should be required to state his choice.
Thereafter formulae may be used at various times as they
arise. Such relations as p = 4a for the perimeter of a square,
or A === a2 for its area, or p = a+b+c for a triangle, I hesitate
to call formulae for the reason that they are too obvious and too
useless. Very seldom does anyone use them as jormulae.
p === a+b+c is, to my mind, a definition written in algebraic
language. To avoid cheapening the word, it ought to be re-
served for more important relations. The formula for con-
verting Fahrenheit temperature into centigrade, and vice versa,
is very useful and desirable as an illustration for negative
numbers except for the difficulties involved in deriving the
formula. There are other problems which not only use formulae
but are interesting in other ways; for example, in every attempt
to correlate mathematics it seems popular to use in algebra
classes the fact that the sum of the angles in a triangle is 180,
and the sum of the angles around a point 360. After working
a few problems involving these two facts I like to ask the class:
"Isn’t the sum of something or other 540 or 720? Why all
this undue emphasis on 180 and 360 always?" We find the
sum of the angles of a square or a rectangle is 360. Is it so
for all four-sided figures? What about five-sided figures? By
 FORMULAS IN THE JUNIOR HIGH SCHOOL 4l3

dividing a polygon into triangles we find the sums of the in-

terior angles and prepare a table showing how the sum of the
angles compares with the number of sides. We write
n, number of sides, 345678
a, sum of angles, 180 360 540 720 900
Without drawing a figure with 369 sides can you predict what
the sum of the angles would be? How do you compute a know-
ing n? We find a = 180(n2). If the pupils do not easily
see the relation from the above table I suggest using a right
angle as the unit of measure so that the numbers for a are
2, 4, 6, 8, 10, ... This formula differs from previous ones in
that it has no meaning if n is a fraction or is less than 3. If the
class then shows sufficient enthusiasm we can consider the
problem of the number of diagonals, d, in an n sided figure. The
sequence of numbers for d is 0, 2, 5, 9, 14, 20, ... By calling
attention to the successive differences, 2, 3, 4, 5, ... of any two
values of d, the pupils easily see how this sequence is built but
some help is needed before they discover that d = n(n3)/2.
These two problems also furnish links in the gap between algebra
and geometry.
The Preliminary Report by the National Committee on Math-
ematical Requirements Mentions
^The formulaits construction, meaning, and use
(a) As a concise language.
(b) As a shorthand rule for computation.
(c) As a general solution.
(d) As an expression of the dependence of one variable on
other variables/7
The items (a), (b), and (c) are well illustrated by the problem
of dividing a line in a certain ratio. Every new type of equa-
tion or quantity affords opportunity for introducing new formu-
lae and further practice on items (a) and (b). But if I under-
stand item (c) correctly, I doubt whether much, time should
be spent on it, especially not in the ninth grade. I believe that
^One of the chief differences between algebra and arithmetic
is that in arithmetic each problem is solved for special numbers,
thus obtaining a result which gives no clew to the result in a
problem entirely similar where different numbers are used.
In algebra letters are used instead of Arabic numerals, thus
obtaining a result that constitutes a formula for the solution
of all problems of the class of which the one solved is a general
Secondary School Circular, No. 6, Bureau of Education. Washington. D. C.
414 SCHOOL SCIENCE AND MATHEMATICS

type. ..... The solution of the general or literal problem

resulting in a formula applicable to all problems of a large class
is essentially the work of algebra. The’ solution of a problem
stated in terms of Arabic numerals is essentially the work of
arithmetic.773 Hence a careful treatment of the item (c) means
an extended study of so-called literal problems, c = lOn is
not a general solution; c = an is. The age problems would
have to be extended until the pupil could solve, ^Find the ages
of A and B if A is now a times as old as B, and h years hence
will be b times as old.57 And even if we do not solve ^A^s rate
exceeds a times B^s rate by b miles per hour; if they walk toward
each other from points c miles apart and A starts e hours before
B, how soon will they meet?77 then item (c) would at least imply
that we solve such problems as ^Find two numbers whose sum
is s and whose difference is d77 or ^If A can do a job in a days
and B do it in b days, how many days are required if they work
together?77 Under the recently proposed reforms in high
school mathematics there would be little time for such general
problems. But there are two general formulae which ought
to be considered, the one for a quadratic, and the other for a
pair of linear simultaneous equations. To develop these in-
telligently and without thrusting them boldly at the pupil
accompanied by an illustration of their use is a difficult matter.
The following method of handling them may be of interest:
After the class has solved simultaneous equations for some
days I assign 5x+9y = 17, 7x6y = 4. When the pupil
has found x = 2, y = 3, I ask, ^If the first equation had con-
tained lOx instead of 5x would not the final value of x have
been different? Does not the pair of answers depend on the
numbers 5, 9, 17, . . in the given equations? Then if I am
told the numbers in the equations, ought I not to be able to
find a rule with which to predict the answers? How might
the rule read? It might say: multiply this number (pointing
to one of the coefficients) by this number perhaps (pointing to
another coefficient) and get 2 for the xV It is an unusual class
in which some one does not say that x can be found by adding
the 7 in the second equation to the 5 in the first, and y can
, .sFrom a pamphlet, "The Teaching of.Algebra, with special reference to Slaught and Lennes
High School Algebra." It is interesting to note that this teixt published in 1908 contains.
many of, the ideas whose value we are beginning to appreciate only todLay. Thus the first
’171 pages deal entirely with the solving of equations, even linear simultaneous equations;
then products of binomials and factoring are treated with immediate applications to equa-
tions. Division of polynomials by polynomials does not arise till page 218 and is then a.
preliminary to square roots, radicals, and quadratic equations. The treatment of fractions
comes last and serves as a general review of equations.
 FORMULAS IN THE JUNIOR HIGH SCHOOL 415

be found by adding the 6 in the second equation to the 9

in the first. I have of course made my equation so that this
will attract his eye, and have deliberately pointed to these
numbers while talking. His method is wrong for surely the 17
and 4 must be taken into consideration. So we begin analyz-
ing our solution. The y = 3 was obtained by dividing a 99
by a 33. Where did the 99 come from? From adding the 119
and the 20. Where did the 119 and 20 come from? From
multiplying the 17 by 7, and from multiplying the 5 by 4
and changing the sign; and these are numbers which appear
in the given equations.
Returning now to the number 33, where did it come from?
Thus we trace our solution backward until we see where and
how every coefficient in the given equations has contributed
to the final denouement. Having thus traced the value for
y I find the pupils get actually excited about tracing the mystery
for x. No youngster ever took a clock apart to see what makes
it tick any more enthusiastically than he dissects the value of
x. Moreover, some of the pupils solved the equations by the
substitution, method and some by the "multiplication and ad-
dition^ method.4 The fact that the same numbers 99, 119,
20, etc., appear in both solutions and that the explanation
of a pupil who used one method applies also to the work of a
pupil who used the other method seems actually miraculous
to the pupils and always causes a great deal of surprise. We
teachers have become so blase to the wonders of mathematics
that we forget many things which can be made to appear won-
derful in the pupiFs eyes.
In order to be able to write a rule in concise language we
decide to call the coefficients a, b, c, exactly as we did in the first
problem about the carpenter and the board. Also we can
derive the general solution directly from the equations ax+by =
+
c, dx ey = /, and the pupils will now understand what is being done.
This same method of explanation can be used in deriving the
formula for the solution of a^+brc+c = 0. But as quadratics
are studied near the end of the year when the pupils have ac-
quired more skill in handling all sorts of literal expressions, the
following shorter method can be used: When the pupil has
^This method is usually called the "addition or subtraction" method. I prefer the name
"multiplication and addition" because the pupil actually does multiply the two equations
by the two numbers and then add. There is never any necessity for subtraction since negative
multipliers may as well be used as positive ones.
 416 SCHOOL SCIENCE AND MATHEMATICS

solved ax^+bx+c = 0 and found x = [-6+v/(&2-4ac)]/2a

I do not call this a formula but develop again the concept by
asking: ^If now you were asked to solve ax2-[-bx-^-d = 0 would
it be correct merely to replace c by d in the solution already
found? Then to solve ax2-{-ex-}-d == 0 could I replace b by e
in my last solution? To solve 3x2-[-ex-}-d == 0 could I replace a
by 3? How can I use my first solution to solve 3x2-[-2x+d = O?
The pupil will then begin to see that one solution containing a,
b, c, is a composite of the solutions of all quadratics. Here I
would like to repeat what was said in a previous paragraph:
that, whenever possible, some of the arithmetic should be per-
formed mentally. The mere sight of an expression [(7)dL
V(-7)2-4(3)(2)]/2.3 as a solution of 3^-7^+2 == 0 is enough
to discourage any pupil, whereas if he is taught to write at once
l7–.^(4Q-24)]/Q the work will be easier.
The object of this paper was to explain how the notion of what
a formula is may be taught. There are teachers who fear that the
use of formulae eliminates thinking and becomes a matter of
formal manipulation, which the pupil may learn to do but never
learn to understand. Their argument seems to hinge on the
assumptions (1) that the pupils in physics or other applied
fields cannot use the mathematics which they have studied,
and (2) that our grandfathers who solved by arithmetic such
problems as ^How much does a fish weigh if the body of the
fish weighs 30 pounces, the head twice as much as the tail, and
the body and head together weigh three times as much as the
head?^ received a training in analytical and independent thinking
that our present system of education doe@ not give the pupil
to-day. ^
All teachers are familiar with the answer to the first objection;
namely, that a pupil may understand a process or method or idea
very well in one set of surroundings but show an amazing unfa-
miliarity with them in some other surroundings. Thus in algebra
the pupil may learn to subtract polynomials and score 100 on
any examination on subtraction but still make various errors
when doing the subtracting necessary in long division. He
may learn to write (x4)2 == x2 8x+lQ but when he solves
2x+(x-^)2 = 5+(^+6)2 he will write ^-16 for (^-4)2. .
Teachers in high school similarly complain that arithmetic is
not taught thoroughly in the grades, forgetting that the pupil
who says 1/2+1/3 == 5/6 in the grades may use 2/5 as the sum
in any unfamiliar situation. A pupil may learn his trigonometry
 FORMULAS IN THE JUNIOR HIGH SCHOOL 417

quite well but not be able to resolve the forces acting on an in-
clined plane into its various components. The cose which he
uses in such a problem is not the GOSO with which he became
acquainted in the trigonometry class because it has different
surroundings. The college student may learn all about E, i,
and R in a physics class but not understand them when he meets
them in a differential equation. Hence good teaching has always
required that every fact should be established by a multiplicity
of associations. And an old idea used in different surroundings
must be carefully developed anew. Physics and chemistry teach-
ers must either avoid formulae in their classes or else take the
time to convince the pupil that the formulae in their texts are
really the same ones that appeared in the algebras.
The second objection refers not only to the teaching of formu-
lae but to all teaching of mathematics, and is more difficult to
refute in a short space. The past has always seemed rosier than
the present. Every adult thinks that the winters were colder
when he was young; that the children in his day had more fun
than the children of to-day, and had to work a great deal harder,
and took life more seriously. And will not the coming generation
say exactly the same things fifty years hence? And who can
answer such impressions? But I think it is safe to say that
almost any youngster of to-day can think more clearly and ana-
lytically than his grandfather did fifty years ago. Just to try
them I copied on the blackboard the data given on page 717 of
SCHOOL SCIENCE AND MATHEMATICS, November, 1920, and asked
what the relation might be between E (the elongation) and P
(the pull). Some of the answers were "E is some numbers times
P/’ "E is about 6 times P,7’ "E == 6P approximately,77 "Draw
the graph for E and P to see whether it^s a line or curve,77 "It
looks as if the graph would be a line,77 "Put E = kP and see if
you can find k.^ Good answers, I think; but I will admit
the class has had frequent exercises in trying to find the relation
between x and y from statistical data as illustrated in the previous
problem about the sum of the angles in a polygon.