Pickett Teaching Guide for Slide Rule Instruction (Manual 601)
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by Maurice L. Hartung
UNIVERSITY of CHICAGO
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Introduction
Many teachers have not had extensive experience in teaching the
Slide rule. They are looking for suggestions for effective ways of
introducing the slide rule to their students. Some teachers who have
had much experience may nevertheless be interested in checking their
own methods against those that others have found effective. This
Teaching Guide has been prepared to help both groups of teachers.
The students who are mentioned in the Guide are assumed to be in
junior or senior high school classes. The Guide is organized into
twelve sections. Some teachers may wish to follow the suggestions in
each of these sections. Others will want to select only certain sec-
tions appropriate for their classes and time allotments. There is also
some freedom of choice as to the order in which the various topics may
be taken up. Comments on decisions as to selections and order are
included in the Guide.
If each teacher who looks through this Guide finds only one sug-
gestion that will help him make the study of mathemetics more in-
teresting and effective for his students, the preparation of the Guide
will have been worth the effort.
Maurice L. Hartung, the author of this guide, received the MA
and PhD degrees in mathematics from the University of Wis-
consin, and is a professor at fhe University of Chicago. An
authority on the teaching of mathematics, he is the author of
many books and articles, and is a qualified expert on the
slide rule.
Copyright 1960
Pickett Ine.
Santa Barbara, California
EEO
Section 1
10
11
12
Contents
Introducing the slide rule... ........2.00 2588004
Helping students understand the basic principles. .......
Teaching multiplication and division ..............
Teaching students about approximations ............
Teaching "scientific notation” ...........0582.80005
Using the ''Powers-of-Ten" slide rule as a learning aid in
teaching exponential notation and decimal point location. ...
Teaching students to use the Aand B scales..........
Teaching students to use the K scale. ...........0..
Teaching students to use the CI scale. ...........-.
Teaching students the meaning of logarithms, how to find
logarithms, and how the L, C, and D scales are made
Teaching the use of the S, T, and ST scales. .........
Teaching slide rule operation in terms of logarithms .....
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Figure 1
Introducing the slide rule
Motivation is no problem in the early stages of teaching students to
use a slide rule. With the instrument in their hands they are ready
and eager to begin using it. Tell the class that the slide rule is a
fairly simple tool by means of which answers to involved mathematical
problems can be easily obtained. To solve the problems easily and
with confidence, however, it is necessary that the students have a
clear understanding of the operation of the slide rule.
Begin by showing the students how to hold a slide rule. The ends of
the rule should be held between thumb and forefinger. This will insure
free, smooth movement of the slide and of the indicator. Holding the
rule at the center hinders this free movement.
The slide rule consists of three parts: (1) body; (2) slide; (3) cursor,
or indicator, with its hairline. The scales on the body and slide are
arranged to work together in solving problems, and each scale is
named by a letter (A, B, C, D, etc.) or other symbol. The hairline
on the indicator is used to help read the scales and adjust the slide.
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CURSOR, OR necieerwLiD 9.)
Introducing the slide rule 5
Helping students t.
get the general idea 1.1 Calli attention to the scales on the slide rule. Tell the students
that the C and D scales are the most important scales, and that the
basic use of these scales is to multiply numbers. Identify the body,
the slide, and the cursor, or indicator, with its hairline. Ifa large
demonstrator rule is available, set it to show 2 x 3 = 6; that is, set
1 on the C scale over 2 on the D scale. Set the hairline of the cursor
over 3 on the C scale and show the class that the answer, 6, may be
read on the D scale under the hairline. Have each student set his own
rule and read the answer from it.
Move the cursor hairline to 4 on the C scale and show that 2 x 4 = 8
may now be read on the D scale.
1.2 Next, write an example such as 2.34 x 36.8 on the board. Tell
the students that before they can do more difficult examples like this,
they must learn how to read the scales and locate the position that cor-
responds to each number.
Helping students Qs
learn to read 2.1 The primary graduations .
the scales 2.11 Tell the students there are a few "vocabulary" terms they
should know to aid in talking about the procedure.
Call attention to the subdivisions of the D scale, and tell the class
that each mark is cailed a graduation.
Figure 2 5 PUMA CUOMO LL CU Ly A
LEFT INDEX | | | | RIGHT INDEX
PRIMARY GRADUATIONS
Set the cursor hairline over the left index of D. Tell the students
this graduation is called the "left-hand index.'' Have them note the
large numeral "1" under it. Next, have them observe the mark at the
right, called the "right-hand index," which is also labeled "1". By
moving the cursor, help the students locate the other major sub-
divisions labeled "2", "3", ''4'', etc. Explain that these ten marks are
called "primary" graduations and represent the first digit in any
numeral. Point out that these primary graduations are closer together
as one moves from left to right. Tell the students that the D scale is
an example of a non-uniform scale.
2.2 The secondary graduations
2.21 Next, have the students place the cursor hairline over the
left-hand index, and move it slowly to the right till they come to the
mark labeled with a small numeral "1", then continue through ''2",
"3" etc., to primary "2".* Tell the students these marks are called
"secondary" graduations. Have the students notice that these gradua-
tions separate the interval between primary "1" and primary "2" into
ten parts. Although these parts do not have the same length--the scale
is non-uniform--the scale is read as though these were tenths. Have
*On 6-inch slide rules these graduations have no numerals beside them.
Figure 3
Figure 4
the students read the sequence between primary "1" and primary "2" as
"1,1, 1.2, 1.3, 1.4, ... 1.9, 2.0." Write these numerals on the
chalkboard in a horizontal line.
Continue to move the cursor hairline to the right over the other
secondary graduations. Have the students read these aloud in sequence
as: "2.1, 2.2, 2.3, ... 3.0, 3.1, 3.2, ... 4.0, 4.1, 4.2, ... 9.8,
9.9, 10.0." Call attention to the fact that in this sequence the right
index is read as 10, not 1.
, MOTT ETI WILLIS a LE eK UU TOU A UC
LEFT INDEX WV RIGHT INDEX
SECONDARY GRADUATIONS
2.22 Now tell the students to move the hairline back to the left index
and start over, but this time they are to read the left index as 10.
Have them read the secondary graduations aloud in sequence, as: "11,
12, 13, ... 20, 21, 22, ... 30, 31, 32, ... 98, 99, 100."" Also write
this sequence of numerals on the chalkboard directly under the others
(i.e., 1.1, 1.2, ...).
2.23 Repeat the cycle with the numerals: "110, 120, 130, ... 200,
210, 220, ... 300, 310, 320, ... 980, 990, 1000." Ask the students to
study these sequences, and draw out for them the generalization that
for each graduation the sequence of digits in the numeral is the same,
but the position of the decimal point is changed.
2.3 The tertiary graduations
2.31 Some teachers may want to defer teaching the students how to
read the third digit until after some work has been done on multiplica-
tion. This is a good plan because it gets the student "right into it"
quickly. However, there are also advantages in going on immediately
to further study of the scales. We assume at this point that the grad-
uation scheme is that customary on 10-inch slide rules.
2.32 Have the students start by placing the hairline over the mark
for 110. Call attention to the short graduations between secondary "1"
and secondary "2". Have these read aloud in sequence as: "111, 112,
113, ... 119, 120.'* Continue with: "121, 122, 123, ... 129, 130, .
198, 199, 200."
2.33 Now start back again at the left index, reading it as "100."
Warn the students that the next reading is "tricky,'’ and help them see
that the sequence is: "101, 102, 103, ... 109, 110."
Move now to primary "2", and call attention to the fact that there
are only five spaces between it and the next secondary graduation. Tell
the students they can now ''count by twos,"' reading the graduations in
sequence as: ''202, 204, 206, 208, 210, 212, ... 396, 398, 400."
149 173 246 247
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LEFT INDEX
Introducing the slide rule 7
2.34 Now have the students start back at 2, and tell them to place
the hairline about halfway between the mark for primary 2 and the next
mark. Tell them this position is read as ''201.'' Move the cursor
between "202" and "204," and read ''203,"" Continue to read the grad-
uations for odd numbers up to 399.
Some sample readings are shown in Figure 5.
101 119 195 223 402 465 694 987
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2.35 A major psychological difficulty may arise at this point. Some
students may be disturbed because they feel they cannot place the hair-
line precisely where it ‘should be." They become unduly concerned
about error. At this time they are not quite ready for a fully satisfy-
ing discussion of this problem. It is best simply to point out that with
practice they will learn to estimate with adequate precision where the
hairline should be. For the present, they should try to place it just
slightly to the right of the midpoint of the interval between two grad-
uations. Point out that engineers, scientists, and others use slide
rules regularly and find the accuracy adequate for much of their work.
If time permits, the students should learn the basic principles of
approximate computation. Some of the essentials will be discussed
later in this teaching guide.
2.36 Next, have the students observe that to the right of primary
"4" there is only one subdivision between each secondary graduation.
Tell the students to read the graduations "by fives," thus: "405, 410,
415,... 505, 510, 515, ... 605, 610, 615, ... 995, 1000.”
Between these graduations the students must estimate. Have them
start at 400 and move the hairline slightly to show where 401 should
be, then move again to 402, 403, 404, and thus arrive at 405. Continue
in this way until they gain some confidence in "'splitting” the interval.
Point out, for example, that 413 will be nearer 415 than 410, and 414
will be very close to 415.
2.37 Finally, by returning once more to the left-hand index, ex-
plain how a fourth digit can be read in the interval between primary "1"
and primary ''2". Begin, for example, at 136, which can be located at
a graduation. By splitting the next interval in half, show where 1365 is
located, Continue with similar examples.
2.38 At this point the students should be given some practice in
reading the C and D scales. The large demonstrator rule may be used
to give them settings to read, and they should each write the numerals
on paper. They should also make settings that correspond to given
numbers. Prolonged practice is not wise, however. The most effective
practice comes with actual computation, where the correct answer
serves as a check on the Settings.
Helping students understand the basic principles
Scales for addition
¢ 3" ~
Some teachers may wish to defer instruction on the basic principles
until after the students have made more progress in learning to compute
with the slide rule. Those teachers may turn at once to Section ITI.
This section may be omitted entirely if only a limited amount
of time is available. If, however, understanding of principles
is a major objective (as it should rightfully be),
time should be found for learning activities of the type described here.
Tell the students that the ordinary slide rule is used to multiply and
divide numbers, and to do many other more complicated computations.
Explain that they will understand better how it works if they take a few
minutes to see how a Slide rule for addition can be made.
1.
1.1 Remind the students that they have used a foot-rule many times
to measure lengths. Demonstrate how two foot-rules, X and Y, can be
used together to add numbers.
For example, in Figure 6 the 0 of Rule X is placed opposite 3 of
Rule Y. Under 4 of Rule X you see 7 of Rule Y. Used in this way,
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Figure 6
The basic principles 9
10
Figure 7
these rules provide a mechanical way of showing that 3+ 4=7. At the
same time, Figure 6 shows that 3+ 5 = 8, and other sums. Also, by
sliding Rule X along Rule Y, the 0 mark of Rule X canbe placed opposite
any mark on Rule Y. The two rulers used together become a slide rule
for simple addition.
1.2 Point out that the marks (or "graduations") and numerals ona
foot-rule are an example of a scale, and that the foot-rule is a uniform
scale. Have the students themselves demonstrate addition using two
ordinary foot-rules, or two uniform scales drawn on paper.
Explain that foot-rules are not often used to add as in Figure 6 be-
cause the scales are short. Only the numbers from 0 to 12 are repre-
sented. Facts such as 3+ 4= 7 are easy to remember, and no one
needs a mechanical device to do such simple examples. Also, have
them notice that the example 3 + 11 causes trouble because the numeral
11 on the X scale is on the part that extends beyond the Y scale. There
is no numeral 14 on a foot-rule.
1.3 Next, demonstrate how examples such as 3 + 11 = 14 and 18 +17
= 35 can be done mechanically by using two yardsticks. The scale ona
yardstick reads from 0 to 36. However, an example, such as 18 + 27,
causes troyble because the sum, 45, is not on the scale of a yardstick.
Bring out through questioning that by using two longer scales this
example can be done. Help students see that any addition example can
be done by using scales that are long enough, but very long scales would
not be convenient to carry around or use. To be convenient, a slide
rule scale should not be much more than about 10 inches in length.
1.4 Now call attention to the fact that the scale on a foot-rule has
marks to represent fractions. Usually the marks show eighths, but
sometimes sixteenths and even finer subdivisions are made. By using
these marks, demonstrate how examples such as
2 3 +6 16 * 8 16
can be done easily with sliding foot-rules. Finally, demonstrate how
marks to represent decimal fractions, such as tenths, can also be used,
as shown on the bottom scale of Figure 7. By using scales graduated
in tenths, show how examples such as 2.3 + 6,9 = 9.2 can easily be done.
Help students see, however, that graduations to show hundredths would
be too close together to read without a magnifier.
1.5 If "meter sticks" are available, as they usually are in science
laboratories, have the students demonstrate addition of numbers
0 1 2 3 4
5 1.5 2.5 3.5
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Scales
for multiplication
expressed to tenths and to hundredths by means of two adjacent meter
sticks.
1.6 Teli the students they will learn how the ordinary slide rule over-
comes the kinds of difficulties mentioned above. The difficulties arise
because the scales must be short. If the scales could be as long as we
pleased some things would be easier, but as a mechanical device the
slide rule would not be the very convenient tool it now is.
2.
2.1 Remind the students again that the scale of a foot-rule is uniform.
The numerals 0, 1, 2, 3, etc., represent consecutive integers (some-
times called "whole numbers"), Ona foot-rule, the distance between
two graduations which represent consecutive integers is always one
inch (see scale L of Figure 8). Have them make a non-uniform scale
by putting marks on a line at intervals of 1 inch and by labeling the
graduations as is done in scale D of Figure 8. The graduation labeled
0 on L is labeled 1 on D; the mark labeled 1 on L is labeled 10 on D;
the mark labeled 2 on L is labeled 100 on D; etc. Tell the students that
the marks for 2, 3, 4, etc., to 9 which would be between 1 and 10 on
scale D need not be shown (see Figure 8). Also, the subdivision marks
for 11, 12, 13, etc., to 99 which would be between 10 and 100 are not
to be showneither. These 89 subdivision marks would have to be put
in the same length (1 inch) as the 8 marks between 1 and10. They
would, therefore, be much closer together than they are on scale L.
Figure 8
0 1 2 3 4 5 é
| | | | | | |
x fl | | | | | |
1 10 100 1000 10,000 100,000 1,000,000
10° 10! 102 103 104 108 106
Similarly, the 899 subdivision marks for integers between 100 and
1000 are not to be shown (see Figure 8). These marks would be very
much closer together than those with the same numerals would be on
scale L. The graduations get more and more crowded as the numbers
get larger. Tell the students scale D is a non-uniform scale, and this
particular non-uniform scale is called a logarithmic scale.
2.2 Explain that these scales may be extended indefinitely. The
only thing that limits the range is the amount of paper or other material
you can use to put the graduations and numerals on. If you had a strip
of paper a mile long, the range of scale L could be from 0 to 63, 360.
The corresponding numbers represented on scale D would then range
from 1 to an enormous number. This number would have 63, 361 digits
The basic principles 11
12
Figure 9
Showing how
short scales
can be used
or figures in its numeral, and it would require a strip of paper about
440 feet long on which to type just this last numeral. Some idea of the
size of this number is gained by remembering that it requires only ten
digits to write "one billion" or 1,000,000,000. Teil the students that a
slide rule which has two logarithmic scales (like scale D of Figure 8)
can be used to multiply one number by another. Explain, for example,
how to find 24 x 380. Use a drawing like Figure 9 on the board. Ex-
plain that you would place the mark for 1 on one of the scales (labeled C
in Figure 9) over the mark for 24 on the other scale. Then you would
look for the mark at 380 on scale C. On scale D right under it you would
find the answer 9, 120.
1) io
cee 100000
2.3 As asummary, make clear that two logarithmic scales, if used
just as the foot-rules were for addition, will provide the answers for
multiplication examples. If the scales are long enough, the accuracy
will be very great, and the position of the decimal point in the answer
numeral can be seen. However, to be practical the scales must be
short--only about 10 inches or less in length. Also, many of the sub-
division graduations (such as for 24 and 38 in Figure 9) cannot be shown,
so the user must learn to estimate where they would be.
3.
3.1 Help the students recall that if we want to measure a distance of
several feet, but have only a foot-rule, we measure by using the same
scale over and over again. That is, we put the scale down and make a
mark to show the first 12 inches, then slide the scale along and use it
again. We repeat this as many times as necessary. In this way we can
measure aS much length as we wish with only a foot-rule, but we must
count and keep track or remember how many times the scale was used.
If necessary, demonstrate this by measuring a length of 3 or more feet
with a foot-rule.
3.2 Then point out that, similarly, if we want to multiply any two
numbers by using logarithmic scales, we can do it with short scales that
show only the numerals from 1 to 10, but we must then keep track of the
decimal point by some other scheme. In reality, we use the same single
section or piece of scale over and over again. When we do this, we in-
terpret the left-hand index as 1, or as 10, or as 100, or as 0.1, or as
0.01, etc., depending upon the number we are setting. Similarly,
primary "2" may represent 2, or 20, or 200, or 0.2, or 0.02, or any
number whose numeral has the sequence of digits below:
...000020000...
We disregard the location of the decimal point in the numeral.
3.3 For example, consider again the problem of finding 24 x 380 by
by using the C and D scales of an ordinary slide rule. First, the left-
hand index of the C scale is set over 24 on the D scale. In this case the
D scale represents the section between 10 and 100 of the complete
logarithmic scale. The numeral "2" on the scale represents 20, and the
numeral '"3'' represents 30, The fourth secondary subdivision between
"2" and "3" represents 24,
Next, the hairline of the cursor is set over 380 on the C scale. In this
case the C scale represents the section between 100 and 1000 of the
complete logarithmic scale.
Help the students see that when the setting is made and read in this
way it is the same as putting down the C scale once and saying “ten,”
then putting it down again and saying "one hundred," and then going on to
380 (see Figure 9). We are now in the section for "hundreds."" The
numeral '"3"' represents 300, and the numeral "4" represents 400. The
eighth secondary subdivision between ''3" and ''4” represents 380.
Now when these settings are made on an ordinary slide rule, the hair-
line of the cursor will be just to the right of the numeral 9 of the D scale.
Since we started at 24 of the D scale, we might think this should repre-
sent 90. However, because two lengths of C scale were used in reaching
the "hundreds," these two lengths must also be accounted for on the D
scale. The numeral 9 near the hairline must now be read as 9000 (see
Figure 9). This example shows how a single section of the logarithmic
scales can be used to get the answers if we know how to interpret the
numerals.
Tell the students that in slide rule computation the decimal point in the
answer is always found mentally or by some auxiliary method. Several
such methods are discussed in a later section.
The basic principles 13
Teaching multiplication and division
14
Multiplication with
the C and D scales
1.
The best procedure is to work out a few examples with the class (using
the demonstrator slide rule) and then help the students generalize by
formulating a ''rule'' such as is given in any manual on slide rule opera-
tion. The first examples should call for the left index of C, and the
result should be readable on D and not be "off scale." The following are
illustrative: 19.5 x 4.1, 2.42 x 296, 121 x 65, 3.81 x 2.16.
1.1 The first basic idea is that in using the ordinary C and D scales,
decimal points in the numerals are ignored.* Thus, 2.34, 23.4,
0.0234, and every number with this sequence of digits is treated as 234.
Remind the class that this is also customary in ordinary multiplication
with paper and pencil.
1.2 The second basic idea the students should have is that multiplica-
tion by slide rule is accomplished by adding two scale lengths and read-~
ing the result on the D scale. (If the suggestions in Section II, parts 1
and 2, have been followed, the students will already know this. If the
suggestions in part 3 have also been used, the students will understand
it still better. )
1.3 The third basic idea is that the location of the decimal point in
the answer is an "extra step," just as it is in paper and pencil work. In
* If “powers of ten” or “decimal keeper” scales are used this statement is not
true. See Section 6.
doing the first few examples, have the students estimate the result by
rounding off the numbers and using mental computation. Encourage
them to make this a habit and use it whenever possible in computation.
The example 19.5 x 4.1 can be rounded to 20 x 4. The students should
see the result is near 80. Tell them they will learn a "scientific"
method of locating the decimal point in a later lesson.
1.4 The fourth basic idea is that if the result falls outside the D scale,
the other index of the C scale should be used. Give the students an
example such as 28.3 x 5.46, and have them set the left index of C over
283 on D. They will find that they cannot move the hairline over 546 of
the C scale. The result is “off the D scale.'' Now have the students set
the right-hand index of C over 28.3 of D and finish the calculation in the
usual way. Make sure they understand that by using one index or the
other, the result will always be on the D scale.
2. Place hairline
over multiplicand (5.46)
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3. Read result (154.5) 1. Set right index of C scale
on D scale. over multiplier (28.3) on D scale.
1.5 If convenient, show the film The Slide Rule I (C and D scales).
1 reel, 16 mm., sound, 24 minutes. B& W. Castle.
1.6 Assign examples for practice, and give individual help to students
who need it.
Division using Bs
the C and D scales First, be sure the students understand that division is the inverse
operation of multiplication. Have them set the slide rule to show 2 x 4 =
8. Point out that we start at 2 on the D scale, then move to 4 on the C
scale, and find the answer 8 under 4 on the D scale. Have them notice
that by reversing these steps we can divide 8 by 4 and get the result 2.
1. Set divisor (4) on C scole
Rene 11 over dividend (8) on D sale
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2. Read result (2) on D scale ff
under left index of C scale
Teaching multiplication and division 15
16
Combined
operations using
the C and D scales
2.1 Point out that the basic idea in dividing by slide rule is to
subtract the length for the divisor from the length for the dividend and
read the result on the D scale.
2.2 Second, tell the students that decimal points are to be ignored in
making the settings and that the location of the decimal point is an extra
step. Have them locate the point by estimating.
2.3 Third, help the students see that in division the result is never
"off scale.'"' Work out examples until most of the students feel confident
that they can practice independently.
3.
To secure more practice immediately and to give students an in-
creased sense of the convenience and power of the slide rule, many
teachers will want to teach how to do ''combined operations" before
systematic methods of locating the decimal point are introduced.
3.1 Best results will be obtained by beginning with examples like
42 x 37
65
First tell the students it is easiest to divide and multiply alternately.
Thus, in the example above we should divide 42 by 65 and then multiply
the result by 37. Help them to see that this means we set (i.e., move)
the slide and the cursor alternately, and that we do not read inter-
mediate results, but only the final answer.
3.2 Second, help the students work out several examples in which
there are more factors; for example:
0.17 x 1.25 x 9. 33 |
4.3 x 6.77
Choose these so there is one more factor in the numerator than in the
denominator.
3.3 Third, use an example such as:
3.72 x 2.46 x 4.31.
2.98
In this example we would set the slide so 298 of C is over 372 of D, then
move the hairline over 2.46 of C.. The result thus far is on D under the
hairline, but we still must multiply by 4.31. To do this we move the
slide so 1 of the C scale is under the hairline, then move the hairline
over 4.31 of C, and read the final result on D. Show the students that
if desired they may write in the factor 1 in the denominator to obtain
3.72 x 2.46 x 4. 31
2.98 x1
and this may help them to remember the settings by emphasizing the
alternate "slide-hairline-slide," etc., movements.
3.4 Finally, show how an example such as 32 x 1.65 x 8.9 can be
done by the same principle if we write (or think)
32 x 1.65 x 8.9 |
Ixi
and again rely on the general principle discussed above.
Helping students
accept
approximations
to numbers
Teaching students about approximations
Students in a modern program of mathematical instruction
should learn the ideas in this section as part of their regular work.
If they have learned these ideas before work in slide rule is introduced,
this section may be omitted. If the students do not already have these ideas,
instruction in them in connection with the slide rule
is not only necessary but also is especially effective.
1.
Many students are reluctant to accept the approximativeness inherent
in much scientific work. They need to give special attention to the
basic ideas involved.
1.1 Explain to the students that much scientific work and other ap-
plications of mathematics involve measurement of some kind. Remind
them that measurement involves error not because of carelessness, but
because of limitations of the measuring instrument. Remind them also
that we can refine our instruments so our results are more precise and
more accurate, but we still will have to accept small errors.
1.2 Tell the students that in slide rule work we are interested in the
accuracy of our results, and that this is indicated by the number of
significant digits we have in the numeral. For example, 704.05 has
five significant digits. Explain that digits other than 0 are always
counted as significant, but sometimes 0 is significant and sometimes
it is not. A 0 between two other digits is always significant. When the
number is less than 1, as for example, 0.00506, the two zeros between
the decimal point and first non-zero digit are not significant. Ina
numeral such as 50600, the two zeros at the right of the 6 may or may
not be significant. However, they are usually counted as not significant
unless there is reason to believe they are significant. For example, a
precise instrument might, in general, give four figures. In that case,
we might know that the underlined figures in 50600 are significant but
the last zero is not.
Teaching approximations 17
18
These ideas should be learned by students in their mathematics
courses whether or not they study slide rules. Give students practice
in counting significant figures.
1.3 Explain that we use approximations to numbers in many cases
even when no measurement is involved. For example, we use 3.14 for
TT, and we use 1. 414 for the square root of 2.
1.4 Tell students that in much scientific and engineering work re-
sults that are accurate to three significant figures are commonly
accepted, at least in preliminary calculations. Remind them that this
accuracy is obtainable with the slide rule--that is one reason it is so
popular, and is taken everywhere as a sort of symbol for engineers
and scientists.
1.5 Explain that when one number that is an approximation is
multiplied or divided by another, the result is no more accurate than
the least accurate of the two numbers. In a paper-and-pencil calcula-
tion of 3. 82 x 6.29 the result, 24.0278, has six digits, but only the
first three, namely 24.0, are to be taken as significant. The slide rule
gives these three significant figures quickly and with no wasted work.
Have them multiply out some examples using first paper and pencil,
then the slide rule, and compare the two methods.
1.6 If time permits, do a few examples such as the following. As-
sume two measurements to the nearest inch are 26 inches and 34 inches.
If new measurements precise to the nearest tenth of an inch were taken,
help the students see that the results might be as large as 26.5 and
34.5, or as small as 25.5 and 33.5. Have the students multiply, by
paper and pencil, 26 x 34 to get 884; then 26.5 x 34.5 to get 914. 25;
then 25.5 x 33.5 to get 854.25. Then point out the original measure-
ments were correct to two significant figures. The result should be
expressed as 880. This comes between 910 and 850, which are the
rounded values of the two extreme values found above. Then show that
the slide rule will also give 880.
Preliminary
comments
Teaching “scientific notation”
Students in a modern program of mathematical instruction
should learn the ideas in this section as part of their regular work.
If they have learned these ideas before work in slide rule is introduced,
this section may be omitted. If the students do not already have these ideas,
instruction in them in connection with the slide rule
is not only necessary but also is especially effective.
1.
The location of the decimal point in the final result has always been
a serious difficulty in slide rule calculation. Many special methods
have been devised, and some of these are explained in most books or
manuals on the slide rule. Most of the special methods involve rules
and auxiliary calculations that are different to learn, remember, and
use. Infact, they are a poor substitute for the one basic or fundamental
method, which uses the "scientific notation" for numbers. In this
notation any real number is expressed, at least approximately, ina
standard form.
The widespread use of this notation in the modern, scientific world
suggests that all students ought to understand this notation. It is there-
fore foolish to waste time teaching most of the special "rules" for
locating the decimal point.
The method of estimating the answer and locating the point is an ex-
ception, and should be taught. Actually, this method amounts to the
mental use of the basic scientific notation. Unfortunately, in some
examples, particularly those involving very large and very small num-
bers, the estimation becomes quite difficult, and for confidence in the
result one must fall back on the basic method.
An effective and simple slide rule method of locating the decimal point
has recently become available. It will give answers in the range 10-10
to 10 10° to one or two significant figures. However, for results more
Teaching "scientific notation" 19
Teaching
the meaning
and value
of exponential
notation
accurate, or outside the range indicated, one must again fall back on
the basic method.
For these reasons instruction should be given in high school on the
use of "scientific notation. '"' In the case of at least some of the brighter
students this work can begin as early as the seventh grade, An under-
standing of "scientific notation" depends upon knowledge of exponents.
2.
2,1 Introduce simple examples involving products of the same factor.
A good example follows.
Number of parents you have is 2.
Number of grandparents is 2x2 = 4.
Number of great-grandparents is 2x2x2 = 8.
In table form:
Number of Generations Back Number of Ancestors
1 2 = 2
2 2x2 =4
3 2x2x2 = 8
4 2xX2xK2x2 = 16
5 2x2x2x2x2 == 32
6 2x2x2x2x2x2 = 64
2.2 Introduce exponent notation as a short way of writing products of
the same factor; thus, write 2 5 instead of 2x2x2x2x2.
2.3 Define terms: e.g., 2 is called the base; 5 is called the
exponent; 32, or 2 3. is called the power. —
2.4 Use as an example the anecdote about the ancient monarch who
wanted to reward the inventor of the game of chess and promised to give
him anything he asked. The inventor said he had a very simple request.
He wanted one kernel of wheat for the first square on the chessboard,
two kernels for the second square, four kernels for the third square,
etc. Write an expression for the number of grains in the 64th square.
(Answer: 2 63 or 2x2x2x2, etc., to 63 factors)
2.5 Use an example with 10 as a base. Have the class prepare a
table as follows:
Power in
Exponent Factors Written Out Exponent Name of Number
Notation
1 10 101 "ten"
2 10x10 102 "one hundred"
3 10x10x10 103 "one thousand"
4 10x10x10x10 104 "ten thousand"
5 10x10x10x10x10 10° "one hundred thousand”
6 10x10x10x10x10x10 10® "one million”
2.6 Summarize to bring out: {a) how exponents make it easy to
write powers of ten; (b) the exponent tells the number of times that 10
Teaching how
to write numbers
in “standard form”
or “scientific
notation”
is used as a factor; (c) for powers of ten the exponent is found by
counting the number of zeros in the repeated factor.
2.7 Assign examples for practice. The following are typical of what
you may use.
Write in exponential notation:
(a) 4x4x4 ¢) 6x6x6x6x6x6x6
(b) 3x3x3 (a) 10x10x10x10x10x10x10
Write the following in the usual way:
(a) 58 (c) 35 (e) 104
(b) 108 (a) 48 (t) 64
3.
3.1 Tell the students that every number, or at least an approximation
to it, can be written in one "standard form." In this form the decimal
point in the numeral is written to the right of the first digit, and then a
factor which is a power of 10 is written. Work out examples such as
the following:
(a) 254 = 2.54x 100 = 2.54 x 102
(b) 683,000 = 6. 83 x 100,000 = 6.83 x 10 9
3.2 Explain the advantages of this way of writing the numerals for
numbers. Among these advantages are the following:
3.21 In many cases itis shorter. Thus, 2.3 x 1013 -
23,000, 000, 000,000. In scientific work the numbers sometimes are
very large or very small.
3.22 When numbers are expressed in this form and then multiplied or
divided, etc., the location of the decimal point in the answer is
simplified.
3.23 This form makes clear just how accurate an approximation one
is dealing with. Thus, we say that 4. 82 is "accurate to three sig-
nificant figures.'' If we see the numeral 482,000, we cannot be sure we
have six significant figures. It is more likely that we still have only
three significant figures. To show this we can write 4.82 x 10° instead
of 482,000. If we actually have six significant figures, we should write
4.82000 x 105. If we have four significant figures, we should write
4.820 x 10°, A recent estimate of the age of the solar system is
4,950, 000, 000 or 4.95 x 109 years. This approximation is certainly
not accurate beyond three significant figures. Help the students see
that this way of showing the accuracy of the result is important in
careful scientific work.
3.3 Explain how to write numerals in "scientific notation." The
easiest method is as follows:
(a) Write the significant digits, a "times" sign (x), and a 10.
(b) Place a decimal point at the right of the first (counted from the
left) non-zero digit.
{c) Start at the right of first non-zero digit in the original numeral
and count the "places"--that is, the digits and zeros passed
over--in reaching the decimal point. The result of the count is
Teaching "scientific notation" 21
Teaching how to use
“scientific notation”
22
to locate
the decimal point
in multiplication
the absolute, or "numerical," value of the exponent. If the
count to the original decimal point moves toward the right, the
exponent is positive (+). If the count moves toward the left, the
exponent is negative (-). Write in the proper symbol for the
exponent of 10.
For example, take 5,790,000. First, write 5.79 x10. Next, start
between the 5 and the 7 and count (toward the right) the six "places."
Write the 6 to indicate the power of 10.
For 0. 000283, first write 2.83 x 10. Now start between the 2 and
the 8 and count (toward the left) the four "places."' The final result is,
then, 2.83 x 1074,
3.4 Show students how to change a numeral from standard form to
ordinary form by reversing the above process. Thus, to express
4.68 x 10° in ordinary form write 468 and then supply zero symbols until
there are 5 places to the right of the 4; the result is 468, 000.
To express 8.93 x 10 a8 in ordinary form write 893 and then su