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3.1
3.1
Measurements and Their Uncertainty
1
Connecting to Your World
On January 4, 2004, the Mars
Exploration Rover Spirit landed on Mars. Equipped with five scientific
instruments and a rock abrasion tool (shown at left), Spirit was sent to
examine the Martian surface around Gusev Crater,
a wide basin that may have once held a lake.
Each day of its mission, Spirit recorded
measurements for analysis. This data helped
scientists learn about the geology and
climate on Mars. All measurements
have some uncertainty. In the chemistry
laboratory, you must strive for accuracy and
precision in your measurements.
Guide for Reading
Key Concepts
• How do measurements relate
to science?
• How do you evaluate accuracy
and precision?
• Why must measurements be
reported to the correct number
of significant figures?
• How does the precision of a
calculated answer compare
to the precision of the
measurements used to
obtain it?
Vocabulary
Using and Expressing Measurements
Your height (67 inches), your weight (134 pounds), and the speed you drive on
the highway (65 miles/hour) are some familiar examples of measurements. A
measurement is a quantity that has both a number and a unit. Everyone
makes and uses measurements. For instance, you decide how to dress in the
morning based on the temperature outside. If you were baking cookies, you
would measure the volumes of the ingredients as indicated in the recipe.
Such everyday situations are similar to those faced by scientists.
Measurements are fundamental to the experimental sciences. For that
reason, it is important to be able to make measurements and to decide
whether a measurement is correct. The units typically used in the sciences
are those of the International System of Measurements (SI).
In chemistry, you will often encounter very large or very small numbers. A single gram of hydrogen, for example, contains approximately
602,000,000,000,000,000,000,000 hydrogen atoms. The mass of an atom of
gold is 0.000 000 000 000 000 000 000 327 gram. Writing and using such
large and small numbers is very cumbersome. You can work more easily
with these numbers by writing them in scientific, or exponential, notation.
In scientific notation, a given number is written as the product of
two numbers: a coefficient and 10 raised to a power. For example, the
number 602,000,000,000,000,000,000,000 written in scientific notation is
6.02 1023. The coefficient in this number is 6.02. In scientific notation, the
coefficient is always a number equal to or greater than one and less than
ten. The power of 10, or exponent, in this example is 23. Figure 3.1 illustrates how to express the number of stars in a galaxy by using scientific
notation. For more practice on writing numbers in scientific notation, refer
to page R56 of Appendix C.
measurement
scientific notation
accuracy
precision
accepted value
experimental value
error
percent error
significant figures
Reading Strategy
Building Vocabulary As you
read, write a definition of each key
term in your own words.
FOCUS
Objectives
3.1.1 Convert measurements to scientific notation.
3.1.2 Distinguish among accuracy,
precision, and error of a measurement.
3.1.3 Determine the number of significant figures in a measurement and in a calculated
answer.
Guide for Reading
Build Vocabulary
Paraphrase Have students write definitions of the words accurate and precise in their own words. As they read
the text, have students compare the
definitions with those of accuracy and
precision given in the text.
Reading Strategy
200,000,000,000. 2 1011
Decimal
moves
11 places
to the left.
Exponent is 11
Figure 3.1 Expressing very large numbers, such as the estimated
number of stars in a galaxy, is easier if scientific notation is used.
Section 3.1 Measurements and Their Uncertainty 63
L2
L2
Use Prior Knowledge Ask, What
everyday activities involve measuring? (Examples include buying consumer products, doing sports activities,
and cooking.) Ask students to recall
which units of measure are related to
each of the examples they give. Have
some students estimate their height in
inches. Have other students measure
the same students with a yardstick or
tape measure. Compare the estimates
with measured values. Ask, What do
you think your height is in centimeters? (Acceptable answers range from
130 to 200 centimeters.)
2
INSTRUCT
Section Resources
Print
• Guided Reading and Study Workbook,
Section 3.1
• Core Teaching Resources, Section 3.1
Review
• Transparencies, T20–T26
Technology
• Interactive Textbook with ChemASAP,
Animation 2, Problem-Solving 3.2, 3.3, 3.6,
3.8, Assessment 3.1
Have students study the photograph and
read the text that opens the section. Ask,
How do you think scientists ensure
measurements are accurate and precise? (Acceptable answers include that scientists make multiple measurements by
using the most precise equipment available. They use samples with known values
to check the reliability of the equipment.)
Scientific Measurement
63
Section 3.1 (continued)
Using and Expressing
Measurements
L1
Use Visuals
Figure 3.1 Have students study the
photograph and read the text that
opens the section. Ask, How is the
exponent of a number expressed in
scientific notation related to the
number of places the decimal point
is moved to the left in a number
larger than 1? (They are equal.)
Figure 3.2 The distribution of
darts illustrates the difference
between accuracy and precision.
a Good accuracy and good
precision: The darts are close to
the bull’s-eye and to one
another. b Poor accuracy and
good precision: The darts are far
from the bull’s-eye but close to
one another. c Poor accuracy
and poor precision: The darts are
far from the bull’s-eye and from
one another.
Poor accuracy
Good precision
Poor accuracy
Poor precision
Accuracy, Precision, and Error
Your success in the chemistry lab and in many of your daily activities
depends on your ability to make reliable measurements. Ideally, measurements should be both correct and reproducible.
Accuracy and Precision Correctness and reproducibility relate to the
concepts of accuracy and precision, two words that mean the same thing to
many people. In chemistry, however, their meanings are quite different.
Accuracy is a measure of how close a measurement comes to the actual or
true value of whatever is measured. Precision is a measure of how close a
series of measurements are to one another.
To evaluate the accuracy
of a measurement, the measured value must be compared to the correct
value. To evaluate the precision of a measurement, you must compare
the values of two or more repeated measurements.
Darts on a dartboard illustrate accuracy and precision in measurement. Let the bull’s-eye of the dartboard represent the true, or correct,
value of what you are measuring. The closeness of a dart to the bull’s-eye
corresponds to the degree of accuracy. The closer it comes to the bull’seye, the more accurately the dart was thrown. The closeness of several
darts to one another corresponds to the degree of precision. The closer
together the darts are, the greater the precision and the reproducibility.
Look at Figure 3.2 as you consider the following outcomes.
Activity
L2
Purpose To illustrate the concepts of
precision and accuracy
Materials a small object (such as a lead
fishing weight), triple-beam balance
Procedure Place the object and a
triple-beam balance in a designated
area. Set a deadline by which each student will have measured the mass of
the object. After everyone has had an
opportunity, have students compile a
summary of all the measurements.
Illustrate precision by having the students find the average and compare
their measurement to it.
Expected Outcome Measured values
should be similar, but not necessarily
identical for all students.
Precision and Accuracy
a. All of the darts land close to the bull’s-eye and to one another. Closeness
to the bull’s-eye means that the degree of accuracy is great. Each dart in
the bull’s-eye corresponds to an accurate measurement of a value.
Closeness of the darts to one another indicates high precision.
b. All of the darts land close to one another but far from the bull’s-eye. The
precision is high because of the closeness of grouping and thus the high
level of reproducibility. The results are inaccurate, however, because of
the distance of the darts from the bull’s-eye.
c. The darts land far from one another and from the bull’s-eye. The results
are both inaccurate and imprecise.
FYI
Other analogies that may be useful in
explaining precision vs. accuracy:
• casting a fishing line
• pitching horseshoes
• a precision marching band
c
b
Good accuracy
Good precision
Accuracy, Precision,
and Error
CLASS
a
Checkpoint How does accuracy differ from precision?
64 Chapter 3
Facts and Figures
Striving for Scientific Accuracy
The French chemist, Antoine Lavoisier,
worked hard to establish the importance of
accurate measurement in scientific inquiry.
Lavoisier devised an experiment to test the
Greek scientists’ idea that when water was
heated, it could turn into earth. For 100 days,
Lavoisier boiled water in a glass flask constructed to allow steam to condense without
64 Chapter 3
escaping. He weighed the water and the
flask separately before and after boiling. He
found that the mass of the water had not
changed. The flask, however, lost a small
mass equal to the sediment he found in the
bottom of it. Lavosier proved that the sediment was not earth, but part of the flask
etched away by the boiling water.
Determining Error Note that an individual measurement may be accurate or inaccurate. Suppose you use a thermometer to measure the boiling
point of pure water at standard pressure. The thermometer reads 99.1°C.
You probably know that the true or accepted value of the boiling point of
pure water under these conditions is actually 100.0°C. There is a difference
between the accepted value, which is the correct value based on reliable
references, and the experimental value, the value measured in the lab. The
difference between the accepted value and the experimental value is called
the error.
Word Origins
Percent comes from the
Latin words per, meaning
“by” or “through,” and
centum, meaning “100.”
What do you think the
phrase per annum
means?
Error experimental value accepted value
Error can be positive or negative depending on whether the experimental
value is greater than or less than the accepted value.
For the boiling-point measurement, the error is 99.1°C 100.0°C, or
0.9° C. The magnitude of the error shows the amount by which the experimental value differs from the accepted value. Often, it is useful to calculate
the relative error, or percent error. The percent error is the absolute value of
the error divided by the accepted value, multiplied by 100%.
Percent error 0error 0
100%
accepted value
Word Origins
L2
The phrase per annum means “by
the year.”
Use Visuals
L1
Figure 3.2 Have students inspect Figure 3.2. Ask, If one of the darts in Figure
3.2c were closer to the bull’s-eye, what
would happen to the accuracy? (The
accuracy would increase.) What would
happen to the precision? (The precision
would increase.) What is the operational
definition of error implied by this figure? (The error is the distance between the
dart and the bull’s-eye.)
L2
Discuss
Review the concept of absolute value.
Ask, What is the meaning of a positive error? (Measured value is greater
than accepted value.) What is the
meaning of a negative error? (Measured value is less than accepted value.)
Explain that the absolute value of the
error is a positive value that describes
the difference between the measured
value and the accepted value, but not
which is greater.
Using the absolute value of the error means that the percent error will
always be a positive value. For the boiling-point measurement, the percent
error is calculated as follows.
099.1C - 100.0C 0
100%
100.0C
0.9C
100%
100.0C
Percent error 0.009 100%
0.9%
Just because a measuring device works doesn’t necessarily mean that it
is accurate. As Figure 3.3 shows, a weighing scale that does not read zero
when nothing is on it is bound to yield error. In order to weigh yourself
accurately, you must first make sure that the scale is zeroed.
Figure 3.3 The scale below has
not been properly zeroed, so the
reading obtained for the person’s
weight is inaccurate. There is a
difference between the person’s
correct weight and the measured
value. Calculating What is the
percent error of a measured
value of 114 lb if the person’s
actual weight is 107 lb?
Section 3.1 Measurements and Their Uncertainty 65
Answers to...
Figure 3.3 7%
Checkpoint
Accuracy is
determined by comparing a measured value to the correct value. Precision is determined by comparing
the values of two or more repeated
measurements.
Scientific Measurement
65
Section 3.1 (continued)
Significant
Significant
FiguresFigures
in Measurements
in Measurements
Supermarkets
Supermarkets
often provide
often
scales
provide
like the
scales
onelike
in Figure
the one3.4.
in Figure
Customers
3.4. Customers
use
use
these scales
these
to measure
scales tothe
measure
weightthe
of produce
weight ofthat
produce
is priced
thatper
is priced
pound.per
If pound. If
you use a scale
you use
thataisscale
calibrated
that is in
calibrated
0.1-lb intervals,
in 0.1-lbyou
intervals,
can easily
you read
can easily
the read the
scale to thescale
nearest
to the
tenth
nearest
of a pound.
tenth ofWith
a pound.
such With
a scale,
such
however,
a scale,you
however,
can you can
also estimate
alsothe
estimate
weight the
to the
weight
nearest
to the
hundredth
nearest hundredth
of a poundof
byanoting
poundthe
by noting the
position ofposition
the pointer
of the
between
pointercalibration
between calibration
marks.
marks.
Suppose you
Suppose
estimate
youa estimate
weight that
a weight
lies between
that lies2.4
between
lb and 2.5
2.4 lb
lb to
and
be2.5 lb to be
2.46 lb. The2.46
number
lb. The
innumber
this estimated
in this estimated
measurement
measurement
has three digits.
has three
Thedigits. The
first two digits
first in
two
the
digits
measurement
in the measurement
(2 and 4) are
(2 known
and 4) are
with
known
certainty.
withBut
certainty. But
the rightmost
the digit
rightmost
(6) has
digit
been
(6)estimated
has been and
estimated
involves
and
some
involves
uncertainty.
some uncertainty.
These threeThese
reported
threedigits
reported
all convey
digits useful
all convey
information,
useful information,
however, and
however,
are and are
called significant
called significant
figures. The
figures.
significant
The significant
figures in afigures
measurement
in a measurement
include include
all of the digits
all ofthat
the are
digits
known,
that are
plus
known,
a last digit
plus athat
lastisdigit
estimated.
that is estimated.
MeaMeasurements surements
must always
must
be reported
always betoreported
the correct
to the
number
correct
of number
significant
of significant
figfigures because
ures
calculated
because answers
calculated
often
answers
depend
often
ondepend
the number
on the
ofnumber
significant
of significant
figures in the
figures
values
inused
the values
in theused
calculation.
in the calculation.
Instruments
Instruments
differ in the
differ
number
in theofnumber
significant
of significant
figures that
figures
can be
that can be
obtained from
obtained
their from
use and
their
thus
usein
and
thethus
precision
in the of
precision
measurements.
of measurements.
The
The
three meterthree
sticks
meter
in Figure
sticks3.5
in can
Figure
be used
3.5 can
to be
make
used
successively
to make successively
more pre- more precise measurements
cise measurements
of the board.
of the board.
Significant Figures in
Measurements
L2
Discuss
Point out that the concept of significant figures applies only to measured
quantities. If students ask why an estimated digit is considered significant,
tell them a significant figure is one that
is known to be reasonably reliable. A
careful estimate fits this definition.
FYI
When calibration marks on an instrument are spaced very close together
(e.g., on certain thermometers and
graduated cylinders), it is sometimes
more practical to estimate a measurement to the nearest half of the smallest
calibrated increment, rather than to
the nearest tenth.
CLASS
Rules forRules
determining
for determining
whether whether
a digit inaadigit
measured
in a measured
value is value
significant:
is significant:
1 Every nonzero
1 Every
digit
nonzero
in a reported
digit in measurement
a reported measurement
is assumed is
toassumed
be
to be
significant. The
significant.
measurements
The measurements
24.7 meters,24.7
0.743meters,
meter,0.743
and 714
meter,
meters
and 714 meters
each express
each
a measure
expressof
a measure
length toofthree
length
significant
to threefigures.
significant figures.
Activity
Olympic Times
L2
Purpose To illustrate how similar
measurements fom different eras
may vary in precision
Materials Almanacs or Internet access
Procedure Have students look up the
winning times for the men’s and
women’s 100-meter dashes at the 1948
and 2000 Olympic Games. Then have
them answer the following question.
Why do the more recently recorded
race times contain more digits to the
right of the decimal? (Because the
technology used for timekeeping
improved to allow for more precise
measurements.)
Expected Outcome Students should
find that the race times from 1948 were
recorded to the nearest tenth of a
second. The race times from 2000 were
recorded to the nearest hundredth of a
second.
66 Chapter 3
Figure 3.4 Figure
The precision
3.4 The
ofprecision
a
of a
weighing scale
weighing
depends
scale
ondepends on
how finely ithow
is calibrated.
finely it is calibrated.
2 Zeros appearing
2 Zerosbetween
appearing
nonzero
between
digits
nonzero
are significant.
digits areThe
significant.
measureThe measurements 7003ments
meters,
7003
40.79
meters,
meters,
40.79
andmeters,
1.503 meters
and 1.503
eachmeters
have four
each have four
significant figures.
significant figures.
Animation Animation
2 See
2 See
how the accuracy
how the
of aaccuracy of a
calculated result
calculated
depends
result depends
on the sensitivity
on theofsensitivity
the
of the
measuring instruments.
measuring instruments.
with ChemASAP
with ChemASAP
3 Leftmost3zeros
Leftmost
appearing
zerosinappearing
front of nonzero
in front digits
of nonzero
are not
digits
significant.
are not significant.
They act as placeholders.
They act as placeholders.
The measurements
The measurements
0.0071 meter,
0.0071
0.42 meter,
meter, 0.42 meter,
and 0.000 099
andmeter
0.000each
099 meter
have only
eachtwo
have
significant
only twofigures.
significant
The figures.
zeros The zeros
to the left are
to not
the left
significant.
are not By
significant.
writing the
By writing
measurements
the measurements
in scien- in scientific notation,
tific
you
notation,
can eliminate
you cansuch
eliminate
placeholding
such placeholding
zeros: in thiszeros:
case,in this case,
1
5
4.23meter,
101 meter,
4.2 10
and
9.9
meter,
10
and
meter.
9.9 105 meter.
7.1 103 meter,
7.1 10
4 Zeros at the
4 Zeros
end of
atathe
number
end ofand
a number
to the right
and to
of the
a decimal
right ofpoint
a decimal
are point are
always significant.
alwaysThe
significant.
measurements
The measurements
43.00 meters,
43.00
1.010
meters,
meters,
1.010
andmeters, and
9.000 meters
9.000
eachmeters
have four
eachsignificant
have fourfigures.
significant figures.
66 Chapter
663 Chapter 3
a
Measured length = 0.6 m
1m
b
Measured length = 0.61 m
10
c
20
30
40
50
60
70
80
90 1m
50
60
70
80
90 1m
Figure 3.5 Three differently
calibrated meter sticks are
used to measure the length
of a board. a A meter stick
calibrated in a 1-m interval.
b A meter stick calibrated in
0.1-m intervals. c A meter stick
calibrated in 0.01-m intervals.
Measuring How many significant figures are reported in each
measurement?
Measured length = 0.607 m
10
20
30
40
5 Zeros at the rightmost end of a measurement that lie to the left of an
understood decimal point are not significant if they serve as placeholders to show the magnitude of the number. The zeros in the
measurements 300 meters, 7000 meters, and 27,210 meters are not
significant. The numbers of significant figures in these values are one,
one, and four, respectively. If such zeros were known measured values,
however, then they would be significant. For example, if all of the
zeros in the measurement 300 meters were significant, writing the
value in scientific notation as 3.00 102 meters makes it clear that
these zeros are significant.
Use Visuals
L1
Figure 3.5 As students inspect Figure
3.5, model the use of meter stick A by
pointing out that one can be certain
that the length of the board is between
0 and 1 m, and one can say that the
actual length is closer to 1 m. Thus, one
can estimate the length as 0.6 m. Similarly, using meter stick B, one can say
with certainty that the length is
between 60 and 70 cm. Because the
length is very close to 60 cm, one
should estimate the length as 61 cm or
0.61 m. Have students study meter
stick C and use similar reasoning to
describe the measurement and estimation process. Ask, If meterstick C were
divided into 0.001-m intervals, as
are most meter sticks, what would
be the estimated length of the board
in meters? (Acceptable answers range
from 0.6065 to 0.6074 m) In millimeters? (606.5 to 607.4 mm)
L2
Discuss
Be sure to review the role of zeros in
determining the number of significant
figures. When adding or subtracting
numbers expressed in scientific notation, remind students that the numbers must all have the same exponent.
6 There are two situations in which numbers have an unlimited number
of significant figures. The first involves counting. If you count
23 people in your classroom, then there are exactly 23 people, and
this value has an unlimited number of significant figures. The second
situation involves exactly defined quantities such as those found
within a system of measurement. When, for example, you write
60 min 1 hr, or 100 cm 1 m, each of these numbers has an unlimited number of significant figures. As you shall soon see, exact quantities do not affect the process of rounding an answer to the correct
number of significant figures.
Section 3.1 Measurements and Their Uncertainty 67
Differentiated Instruction
L1
Have students write in their own words the
rules for determining the number of significant digits. Help them if necessary. Have
them measure several lengths, masses, and
volumes; then have them use their rules to
determine the correct number of significant
figures for each measurement.
Less Proficient Readers
Answers to...
Figure 3.5 The measurements are:
a. 0.6 m (1 significant figure), b. 0.61
m (2 significant figures), c. 0.607 m
(3 significant figures)
Scientific Measurement
67
Section 3.1 (continued)
CLASS
CONCEPTUAL PROBLEM 3.1
Counting Significant Figures in Measurements
Activity
L2
Purpose To provide practice in applying the rules governing the significance of zeros in measurements
Materials textbook and scientific literature, index cards
Procedure Have students search their
textbooks and other sources for
length, mass, volume, or temperature
measurements that contain zeros.
Have them include some examples
written in scientific notation. Ask them
to write each measurement on the
front of an index card; on the back of
each card, have them write (1) all the
rules governing the significance of
zeros that apply to the measurement,
and (2) the number of significant figures in the measurement. Have pairs of
students exchange index cards and
agree on the appropriateness of the
rules and the answers.
Expected Outcome Students should
be able to apply correctly rules 2–5
listed on pages 66 and 67.
Significant Zeros
How many significant figures are in each measurement?
a. 123 m
b. 40,506 mm
c. 9.8000 104 m
d. 22 meter sticks
e. 0.070 80 m
f. 98,000 m
Analyze Identify the relevant concepts.
Solve Apply the concepts to this problem.
The location of each zero in the measurement
and the location of the decimal point determine which of the rules apply for determining
significant figures.
All nonzero digits are significant (rule 1). Use
rules 2 through 6 to determine if the zeros are
significant.
a. three (rule 1)
b. five (rule 2)
c. five (rule 4)
d. unlimited (rule 6)
e. four (rules 2, 3, 4)
f. two (rule 5)
Practice Problems
1.Practice
Count theProblems
significant figures in each length.
a. 0.057 30 meters
c. 0.000 73 meters
2. How many significant figures are in each
measurement?
a. 143 grams
c. 8.750 102 gram
with ChemASAP
Suppose you use a calculator to find the area of a floor that measures
7.7 meters by 5.4 meters. The calculator would give an answer of
41.58 square meters. The calculated area is expressed to four significant figures. However, each of the measurements used in the calculation is
expressed to only two significant figures. So the answer must also be
In general, a calculated
reported to two significant figures (42 m2).
answer cannot be more precise than the least precise measurement from
which it was calculated. The calculated value must be rounded to make it
consistent with the measurements from which it was calculated.
Rounding To round a number, you must first decide how many signifi-
Answers
cant figures the answer should have. This decision depends on the given
measurements and on the mathematical process used to arrive at the
answer. Once you know the number of significant figures your answer
should have, round to that many digits, counting from the left. If the digit
immediately to the right of the last significant digit is less than 5, it is
simply dropped and the value of the last significant digit stays the same. If
the digit in question is 5 or greater, the value of the digit in the last significant place is increased by 1.
1. a. 4 b. 4 c. 2 d. 5
2. a. 3 b. 2 c. 4 d. 4
L2
Chapter 3 Assessment problem 58 is
related to Sample Problem 3.1.
FYI
Due to rounding, there will often be
discrepancies between actual values
and calculated values derived from
measurements. Some students may
conclude (correctly) that following the
rules for significant figures introduces
error in calculated values. Such rounding errors are generally small, but
should nonetheless be acknowledged
when performing calculations.
68 Chapter 3
b. 0.074 meter
d. 1.072 meter
Problem-Solving 3.2 Solve
Problem 2 with the help of an
interactive guided tutorial.
Significant Figures in Calculations
CONCEPTUAL PROBLEM 3.1
Practice Problems Plus
b. 8765 meters
d. 40.007 meters
Checkpoint Why must a calculated answer generally be rounded?
68 Chapter 3
Significant Figures in
Calculations
Sample Problem 3.1
SAMPLE PROBLEM 3.1
Answers
Rounding Measurements
3. a. 8.71 × 101 m
b. 4.36 × 108 m
c. 1.55 × 10–2 m
d. 9.01 × 103 m
e. 1.78 × 10–3 m
f. 6.30 × 102 m
4. a. 9 × 101 m
b. 4 × 108 m
c. 2 × 10–2 m
d. 9 × 103 m
e. 2 × 10–3 m
f. 6 × 102 m
Round off each measurement to the number of significant figures
shown in parentheses. Write the answers in scientific notation.
a. 314.721 meters (four)
b. 0.001 775 meter (two)
c. 8792 meters (two)
Analyze Identify the relevant concepts.
Round off each measurement to the number of significant figures
indicated. Then apply the rules for expressing numbers in scientific
notation.
Solve Apply the concepts to this problem.
Count from the left and apply the rule to the digit immediately to the
right of the digit to which you are rounding. The arrow points to the
digit immediately following the last significant digit.
a. 314.721 meters
c
2 is less than 5, so you do not round up.
314.7 meters 3.147 102 meters
b. 0.001 775 meter
c
7 is greater than 5, so round up.
0.0018 meter 1.8 103 meter
c. 8792 meters
c
9 is greater than 5, so round up
8800 meters 8.8 103 meters
Practice Problems Plus
Math
Handbook
For help with scientific
notation, go to page R56.
Practice Problems
Evaluate Do the results make sense?
The rules for rounding and for writing numbers in scientific notation
have been correctly applied.
3. Round each measurement to
three significant figures. Write
your answers in scientific
notation.
a. 87.073 meters
b. 4.3621 108 meters
c. 0.01552 meter
d. 9009 meters
e. 1.7777 103 meter
f. 629.55 meters
4. Round each measurement in
Practice Problem 3 to one significant figure. Write each of
your answers in scientific
notation.
Round each measurement to two significant figures. Write your answers in
scientific notation.
a. 94.592 grams (9.5 × 101g)
b. 2.4232 × 103 grams (2.4 × 103g)
c. 0.007 438 grams (7.4 × 10–3g)
d. 54 752 grams (5.5 × 104g)
e. 6.0289 × 10–3 grams (6.0 × 10–3 g)
f. 405.11 grams (4.1 × 102g)
Math
Practice Problems
Problem-Solving 3.3 Solve
Problem 3 with the help of an
interactive guided tutorial.
L2
Handbook
For a math refresher and practice,
direct students to scientific notation,
page R56.
with ChemASAP
Section 3.1 Measurements and Their Uncertainty 69
Answers to...
Checkpoint
A calculated
answer must be rounded in order to
make it consistent with the
measurements from which it was
calculated. The calculated answer
cannot be more precise than the
least precise measurement used in
the calculation.
Scientific Measurement
69
Section 3.1 (continued)
Addition and Subtraction The answer to an addition or subtraction
calculation should be rounded to the same number of decimal places (not
digits) as the measurement with the least number of decimal places. Work
through Sample Problem 3.2 below which provides an example of rounding
in an addition calculation.
L2
Discuss
The rules for rounding calculated numbers can be compared with the old
adage,“A chain is only as strong as its
weakest link.” Explain that an answer
cannot be more precise than the least
precise value used to calculate the
answer. Ask, In addition and subtraction, what is the least precise value?
(The measurement with the fewest digits
to the right of the decimal point.) In multiplication and division, what is the
least precise value? (The measurement
with the fewest significant figures.) If students wonder why addition and subtraction rules differ from multiplication and
division rules, point out that in addition
and subtraction of measurements, the
measurements are of the same property,
such as length or volume. However, in
the multiplication and division of measurements, new quantities or properties
are being described, such as speed
(length ÷ time), area (length × length),
and density (mass ÷ volume).
SAMPLE PROBLEM 3.2
Significant Figures in Addition
Calculate the sum of the three measurements. Give the answer to the
correct number of significant figures.
12.52 meters ⫹ 349.0 meters ⫹ 8.24 meters
Analyze Identify the relevant concepts.
Calculate the sum and then analyze each measurement to determine
the number of decimal places required in the answer.
Solve Apply the concepts to this problem.
Align the decimal points and add the numbers. Round the answer to
match the measurement with the least number of decimal places.
Math
Handbook
For help with significant
figures, go to page R59.
369.76
meters
The second measurement (349.0 meters) has the least number of digits
(one) to the right of the decimal point. Thus the answer must be
rounded to one digit after the decimal point. The answer is rounded to
369.8 meters, or 3.698 ⫻ 102 meters.
Practice Problems
Answers
5. Perform each operation.
5. a. 79.2 m b. 7.33 m c. 11.53 m
d. 17.3 m
6. 23.8 g
Practice Problems Plus
L2
Problem-Solving 3.6 Solve
Problem 6 with the help of an
interactive guided tutorial.
with ChemASAP
Find the total mass of four stones
with the following masses: 10.32
grams, 11.81 grams, 124.678 grams,
and 0.9129 gram. (147.72 g)
Express your answers to the
correct number of significant
figures.
a. 61.2 meters ⫹ 9.35 meters ⫹
8.6 meters
b. 9.44 meters ⫺ 2.11 meters
c. 1.36 meters ⫹ 10.17 meters
d. 34.61 meters ⫺ 17.3 meters
6. Find the total mass of three
diamonds that have masses of
14.2 grams, 8.73 grams, and
0.912 gram.
Handbook
For a math refresher and review,
direct students to scientific notation,
page R56.
70 Chapter 3
Sample Problem 3.3
Math
Answers
Practice Problems Plus
Handbook
For a math refresher and practice,
direct students to using a calculator,
page R62.
7. a. 1.8 × 101 m2 b. 6.75 × 102 m
c. 5.87 × 10–1 min
8. 1.3 × 103 m3
L2
Calculate the volume of a house that has
dimensions of 12.52 meters by 36.86
meters by 2.46 meters. (1.14 × 103 m3)
70 Chapter 3
meters
meters
meters
Evaluate Does the result make sense?
The mathematical operation has been correctly carried out and the
resulting answer is reported to the correct number of decimal places.
Sample Problem 3.2
Math
12.52
349.0
+ 8.24
Multiplication and Division In calculations involving multiplication
Quick LAB
and division, you need to round the answer to the same number of significant figures as the measurement with the least number of significant figures. The position of the decimal point has nothing to do with the rounding
process when multiplying and dividing measurements. The position of the
decimal point is important only in rounding the answers of addition or
subtraction problems.
L2
Objectives After completing this activity, students will be able to
• measure length with accuracy and
precision.
• apply rules for rounding answers calculated from measurements.
• determine experimental error and
express it as percent error.
Accuracy and Precision
Checkpoint How many significant figures must you round an answer to
when performing multiplication or division?
SAMPLE PROBLEM 3.3
Significant Figures in Multiplication and Division
Perform the following operations. Give the answers to the correct
number of significant figures.
a. 7.55 meters ⫻ 0.34 meter
b. 2.10 meters ⫻ 0.70 meter
c. 2.4526 meters ⫼ 8.4
Analyze Identify the relevant concepts.
Students may contend that making
one measurement of some property,
such as length, is satisfactory. Ask,
What possible errors may occur
when making only one length measurement? (Acceptable answers include
misreading the ruler or not holding the
ruler parallel to the length of the object.)
Perform the required math operation and then analyze each of the
original numbers to determine the correct number of significant
figures required in the answer.
Solve Apply the concepts to this problem.
Round the answers to match the measurement with the least number
of significant figures.
a. 7.55 meters ⫻ 0.34 meter ⫽ 2.567 (meter)2 ⫽ 2.6 meters2
(0.34 meter has two significant figures)
b. 2.10 meters ⫻ 0.70 meter ⫽ 1.47 (meter)2 ⫽ 1.5 meters2
(0.70 meter has two significant figures)
c. 2.4526 meters ⫼ 8.4 ⫽ 0.291 976 meter ⫽ 0.29 meter
(8.4 has two significant figures)
Math
Handbook
For help with using a
calculator, go to page R62.
Skills Focus Measuring, calculating
Prep Time 5 minutes
Materials 3 inch × 5 inch index cards,
metric rulers
Class Time 15 minutes
Evaluate Do the results make sense?
The mathematical operations have been performed correctly, and the
resulting answers are reported to the correct number of places.
Teaching Tips
Practice Problems
7. Solve each problem. Give your
answers to the correct number of significant figures and
in scientific notation.
a. 8.3 meters ⫻ 2.22 meters
b. 8432 meters ⫼ 12.5
c. 35.2 seconds ⫻ 1 minute
60 seconds
8. Calculate the volume of a
warehouse that has inside
dimensions of 22.4 meters by
11.3 meters by 5.2 meters.
(Volume ⫽ l ⫻ w ⫻ h)
Problem-Solving 3.8 Solve
Problem 8 with the help of an
interactive guided tutorial.
with ChemASAP
Emphasize that students should use an
interior, marked line, such as 10.0 cm, as
the initial point, instead of the end of the
ruler, which may be damaged.
Expected Outcome Measured values
should be similar, but not necessarily
identical for all students.
Analyze and Conclude
71
For Enrichment
L3
Have students devise methods of calculating
the volume of one card. Point out that measuring the thickness of one card with a ruler would
be very inaccurate. Ask, How might the measurement of the thickness of the card be
improved? (Use a more precise instrument, such
as a micrometer, or measure the thickness of a
number of cards and divide by the number of
cards.) Have students determine the thickness
of one card and calculate its volume. Using the
class average of the calculated volumes, have
each student determine the percent error
using the average as the accepted value.
1. Four for length; three for width
2. See Expected Outcome.
3. Significant digits for rounded-off
answers are area, 3, and perimeter, 4.
Some students may not round to the
proper number of digits.
4. Errors of ±0.03 cm are acceptable.
Such errors yield percent errors of
0.2% for length and 0.4% for width.
Answers to...
Checkpoint
The same number of significant figures as the measurement with the least number of
significant figures.
Scientific Measurement
71
Quick LAB
Section 3.1 (continued)
3
Accuracy and Precision
ASSESS
Evaluate Understanding
L2
Write the following sets of measurements on the board.
(1) 78°C, 76°C, 75°C
(2) 77°C, 78°C, 78°C
(3) 80°C, 81°C, 82°C
Ask, If these sets of measurements
were made of the boiling point of a
liquid under similar conditions,
explain which set is the most precise? (Set 2 is the most precise because
the three measurements are closest
together.) What would have to be
known to determine which set is the
most accurate? (the accepted value of
the liquid’s boiling point)
Reteach
Purpose
Procedure
To measure the dimensions
of an object as accurately
and precisely as possible
and to apply rules for rounding answers calculated
from the measurements.
1. Use a metric ruler to measure in centimeters the length and width of an
index card as accurately and precisely
as you can. The hundredths place
in your measurement should be
estimated.
Materials
3 inch 5 inch index
• card
• metric ruler
2. Calculate the perimeter [2 (length width)] and the area (length width)
of the index card. Write both your
unrounded answers and your correctly
rounded answers on the chalkboard.
Analyze and Conclude
1. How many significant figures are in
your measurements of length and
of width?
2. How do your measurements compare
with those of your classmates?
3. How many significant figures are in
your calculated value for the area? In
your calculated value for the perimeter? Do your rounded answers have as
many significant figures as your classmates’ measurements?
4. Assume that the correct (accurate)
length and width of the card are
12.70 cm and 7.62 cm, respectively.
Calculate the percent error for each of
your two measurements.
L1
Use Figure 3.5 to reteach the method
of correctly recording the number of
significant figures in a measurement.
Then have students convert each measurement into scientific notation.
(6 × 10–1 m, 6.1 × 10–1 m, 6.07 × 10–1 m)
Acceptable answers will include the
following information: Accuracy
compares a measured value to an
accepted value of the measurement,
precision compares a measured value
to a set of measurements made under
similar conditions, and error is the
difference between the measured and
accepzzted values.
3.1 Section Assessment
9.
Key Concept How do measurements relate to
experimental science?
10.
Key Concept How are accuracy and precision
evaluated?
11.
Key Concept Why must a given measurement
always be reported to the correct number of significant figures?
12.
Key Concept How does the precision of a calculated answer compare to the precision of the
measurements used to obtain it?
13. A technician experimentally determined the
boiling point of octane to be 124.1°C. The actual
boiling point of octane is 125.7°C. Calculate the
error and the percent error.
15. Solve the following and express each answer in
scientific notation and to the correct number of
significant figures.
a. (5.3 104) (1.3 104)
b. (7.2 104) (1.8 103)
c. 104 103 106
d. (9.12 101) (4.7 102)
e. (5.4 104) (3.5 109)
Explanatory Paragraph Explain the differences
between the accuracy, precision, and error of a
measurement.
14. Determine the number of significant figures in
each of the following.
a. 11 soccer players
b. 0.070 020 meter
c. 10,800 meters
d. 5.00 cubic meters
Assessment 3.1 Test yourself
on the concepts in Section 3.1.
with ChemASAP
72 Chapter 3
If your class subscribes to the
Interactive Textbook, use it to
review key concepts in Section 3.1.
with ChemASAP
72 Chapter 3
Section 3.1 Assessment
9. Making correct measurements is fundamental to the experimental sciences.
10. Accuracy is the measured value compared
to the correct values. Precision is comparing more than one measurement.
11. The significant figures in a calculated
answer often depend on the number of
significant figures of the measurements
used in the calculation.
12. A calculated answer cannot be more precise than the least precise measurement
used in the calculation.
13. error = –1.6°C; percent error = 1.3%
14. a. unlimited b. 5 c. 3 d. 3
15. a. 6.6 × 104 b. 4.0 × 10–7 c. 107
d. 8.65 × 10–1 e. 1.9 × 1014