Objectives
• Perform calculations involving scientific
notation and conversion factors.
• Identify the metric and SI units used in science
and convert between common metric
prefixes.
• Compare and contrast accuracy and precision.
• Relate the Celsius, Kelvin, and Fahrenheit
temperature scales.
1.3 Measurement
How old are you? How tall
are you? The answers to
these questions are
measurements.
Measurements are
important in both science
and everyday life. It would
be difficult to imagine
doing science without any
measurements.
1.3 Measurement
Using Scientific Notation
Why is scientific notation useful?
1.3 Measurement
Using Scientific Notation
Why is scientific notation useful?
Scientists often work with
very large or very small
numbers. Astronomers
estimate there are
200,000,000,000 stars in
our galaxy.
1.3 Measurement
Using Scientific Notation
Scientific notation is a way of expressing a
value as the product of a number between 1
and 10 and a power of 10.
For example, the speed of light is about
300,000,000 meters per second. In scientific
notation, that speed is 3.0 × 108 m/s. The
exponent, 8, tells you that the decimal point is
really 8 places to the right of the 3.
1.3 Measurement
Using Scientific Notation
For numbers less than 1 that are written in
scientific notation, the exponent is negative.
For example, an average snail’s pace is
0.00086 meters per second. In scientific
notation, that speed is 8.6 × 10-4 m/s.
The negative exponent tells you how many
decimals places there are to the left of the 8.6.
1.3 Measurement
Using Scientific Notation
To multiply numbers written in scientific
notation, you multiply the numbers that
appear before the multiplication signs and
add the exponents. The following example
demonstrates how to calculate the distance
light travels in 500 seconds.
This is about the distance between the sun
and Earth.
1.3 Measurement
Using Scientific Notation
When dividing numbers written in
scientific notation, you divide the numbers
that appear before the exponential terms
and subtract the exponents. The following
example demonstrates how to calculate
the time it takes light from the sun to
reach Earth.
1.3 Measurement
Using Scientific Notation
Using Scientific Notation
A rectangular parking lot has a length of
1.1 × 103 meters and a width of 2.4 ×
103 meters. What is the area of the
parking lot?
1.3 Measurement
Using Scientific Notation
Read and Understand
What information are you given?
1.3 Measurement
Using Scientific Notation
Read and Understand
What information are you given?
1.3 Measurement
Using Scientific Notation
Plan and Solve
What unknown are you trying to calculate?
What formula contains the given quantities and the
unknown?
Replace each variable with its known value
1.3 Measurement
Using Scientific Notation
Look Back and Check
Is your answer reasonable?
Yes, the number calculated is the product
of the numbers given, and the units (m2)
indicate area.
1.3 Measurement
Using Scientific Notation
Look Back and Check
1. Perform the following calculations.
Express your answers in scientific
notation.
a. (7.6 × 10-4 m) × (1.5 × 107 m)
b. 0.00053 ÷ 29
1.3 Measurement
Let’s Practice
• Scientific Notation – as a class
• On Your Own – Practice Exercises
1.3 Measurement
SI Units of Measurement
What units do scientists use for their
measurements?
1.3 Measurement
SI Units of Measurement
Scientists use a set of measuring units
called SI, or the International System of
Units.
• SI is an abbreviation for Système International
d’Unités.
• SI is a revised version of the metric system,
originally developed in France in 1791.
• Scientists around the world use the same
system of measurements so that they can
readily interpret one another’s measurements.
1.3 Measurement
SI Units of Measurement
If you told one of your friends
that you had finished an
assignment “in five,” it could
mean five minutes or five
hours. Always express
measurements in numbers
and units so that their
meaning is clear.
These students’ temperature
measurement will include a
number and the unit, °C.
1.3 Measurement
SI Units of Measurement
Base Units and Derived Units
SI is built upon seven metric units, known as
base units.
• In SI, the base unit for length, or the straightline distance between two points, is the meter
(m).
• The base unit for mass, or the quantity of matter
in an object or sample, is the kilogram (kg).
1.3 Measurement
SI Units of Measurement
Seven metric base units make up the
foundation of SI.
1.3 Measurement
SI Units of Measurement
Additional SI units, called derived units,
are made from combinations of base
units.
• Volume is the amount of space taken up
by an object.
• Density is the ratio of an object’s mass
to its volume:
1.3 Measurement
SI Units of Measurement
Specific combinations of SI base units yield
derived units.
1.3 Measurement
SI Units of Measurement
To derive the SI unit for density,
you can divide the base unit for
mass by the derived unit for
volume. Dividing kilograms by
cubic meters yields the SI unit for
density, kilograms per cubic meter
(kg/m3).
A bar of gold has more mass per
unit volume than a feather, so gold
has a greater density than a
feather.
1.3 Measurement
SI Units of Measurement
Metric Prefixes
The metric unit is not always a convenient one to
use. A metric prefix indicates how many times a unit
should be multiplied or divided by 10.
Metric Conversions
Ladder Method
T. Trimpe 2008 http://sciencespot.net/
Ladder Method
1
2
KILO
1000
Units
3
HECTO
100
Units
DEKA
10
Units
DECI
0.1
Unit
Meters
Liters
Grams
How do you use the “ladder” method?
CENTI
0.01
Unit
MILLI
0.001
Unit
4 km = _________ m
1st – Determine your starting point.
Starting Point
2nd – Count the “jumps” to your ending point.
How many jumps does it take?
3rd – Move the decimal the same number of
jumps in the same direction.
Ending Point
4. __. __. __. = 4000 m
1
2
3
Conversion Practice
Try these conversions using the ladder method.
1000 mg = _______ g
1 L = _______ mL
160 cm = _______ mm
14 km = _______ m
109 g = _______ kg
250 m = _______ km
Compare using <, >, or =.
56 cm
6m
7g
698 mg
Metric Conversion Challenge (worksheet)
Write the correct abbreviation for each metric unit.
1) Kilogram _____
4) Milliliter _____
7) Kilometer _____
2) Meter _____
5) Millimeter _____
8) Centimeter _____
3) Gram _____
6) Liter _____
9) Milligram _____
Try these conversions, using the ladder method.
10) 2000 mg = _______ g
15) 5 L = _______ mL
20) 16 cm = _______ mm
11) 104 km = _______ m
16) 198 g = _______ kg
21) 2500 m = _______ km
12) 480 cm = _____ m
17) 75 mL = _____ L
22) 65 g = _____ mg
13) 5.6 kg = _____ g
18) 50 cm = _____ m
23) 6.3 cm = _____ mm
14) 8 mm = _____ cm
19) 5.6 m = _____ cm
24) 120 mg = _____ g
Compare using <, >, or =.
25) 63 cm
26) 536 cm
6m
53.6 dm
27) 5 g
28) 43 mg
508 mg
5g
29) 1,500 mL
30) 3.6 m
1.5 L
36 cm
Objectives
• Perform calculations involving scientific
notation and conversion factors.
• Identify the metric and SI units used in science
and convert between common metric
prefixes.
• Compare and contrast accuracy and precision.
• Relate the Celsius, Kelvin, and Fahrenheit
temperature scales.
1.3 Measurement
SI Units of Measurement
For example, the time it takes for a
computer hard drive to read or write
data is in the range of thousandths of a
second, such as 0.009 second. Using
the prefix milli- (m), you can write 0.009
second as 9 milliseconds, or 9 ms.
1.3 Measurement
SI Units of Measurement
A conversion factor is a ratio of equivalent
measurements used to convert a quantity expressed
in one unit to another unit.
To convert the height of Mount Everest, 8848
meters, into kilometers, multiply by the conversion
factor on the left.
1.3 Measurement
SI Units of Measurement
To convert 8.848 kilometers back into meters,
multiply by the conversion factor on the right. Since
you are converting from kilometers to meters, the
number should get larger.
In this case, the kilometer units cancel, leaving you
with meters.
Conversion Factors
Fractions in which the numerator and denominator are
EQUAL quantities expressed in different units
Example:
1 in. = 2.54 cm
Factors: 1 in.
and 2.54 cm
2.54 cm
1 in.
How many minutes are in 2.5 hours?
Conversion factor
2.5 hr x 60 min
1 hr
= 150 min
cancel
By using dimensional analysis / factor-label method, the UNITS
ensure that you have the conversion right side up, and the UNITS
are calculated as well as the numbers!
Sample Problem
• You have $7.25 in your pocket in quarters.
How many quarters do you have?
7.25 dollars X 4 quarters
1 dollar
= 29 quarters
Learning Check
A rattlesnake is 2.44 m long. How long is the
snake in cm?
a) 2440 cm
b) 244 cm
c) 24.4 cm
Solution
A rattlesnake is 2.44 m long. How long is the
snake in cm?
b) 244 cm
2.44 m x 100 cm
1m
= 244 cm
Learning Check
How many seconds are in 1.4 days?
Unit plan: days
hr
min
seconds
1.4 days x 24 hr x60 min x 60 s
1 hr
1 min
1 day
= 1.2 x 105 s
Wait a minute!
What is wrong with the following setup?
1.4 day
x 1 day
24 hr
x
60 min x 60 sec
1 hr
1 min
English and Metric Conversions
• If you know ONE conversion for each type of
measurement, you can convert anything!
• I will provide these equalities, but you must
be able to use them:
– Mass: 454 grams = 1 pound
– Length: 2.54 cm = 1 inch
– Volume: 0.946 L = 1 quart
Steps to Problem Solving

Read problem
 Identify data
 Make a unit plan from the initial unit to the desired
unit (good practice at beginning, not necessary as you
get comfortable with this)
 Select conversion factors
 Change initial unit to desired unit
 Cancel units and check
 Do math on calculator
 Give an answer using significant figures
Dealing with Two Units
If your pace on a treadmill is 65 meters per minute,
how many seconds will it take for you to walk a
distance of 8450 feet?
HINT: Always start with the simplest label.
You’re looking for seconds, so you can’t start there.
65 m/min has two labels so that’s not very simple.
Best STARTING place is 8450 feet!
What about Square and Cubic units?
• Use the conversion factors you already know,
but when you square or cube the unit, don’t
forget to cube the number also!
• Best way: Square or cube the ENTIRE
conversion factor
• Example: Convert 4.3 cm3 to mm3
4.3 cm3
( )
10 mm
1 cm
3
=
4.3 cm3
103 mm3
13 cm3
= 4300 mm3
Learning Check
• A Nalgene water
bottle holds 1000
cm3 of dihydrogen
monoxide (DHMO).
How many cubic
decimeters is that?
Solution
1000 cm3
1 dm 3
10 cm
( )
= 1 dm3
So, a dm3 is the same as a Liter !
A cm3 is the same as a milliliter.
Let’s Practice
• Dimensional Analysis Notes
• Dimensional Analysis Worksheet 1
• Find the percentage error.
Objectives
• Compare and contrast accuracy and precision.
• Utilize the rules of significant figures.
• Relate the Celsius, Kelvin, and Fahrenheit
temperature scales.
1.3 Measurement
Limits of Measurement
How does the precision of
measurements affect the precision of
scientific calculations?
1.3 Measurement
Limits of Measurement
Precision
Precision is a gauge of how exact a
measurement is.
Significant figures are all the digits that
are known in a measurement, plus the last
digit that is estimated.
1.3 Measurement
Limits of Measurement
The precision of a calculated answer is
limited by the least precise
measurement used in the calculation.
1.3 Measurement
Limits of Measurement
A more precise time can be read from the digital clock than
can be read from the analog clock. The digital clock is
precise to the nearest second, while the analog clock is
precise to the nearest minute.
Significant Figures
► When using our calculators we must determine the correct
answer; our calculators are mindless drones and don’t know
the correct answer.
► There are 2 different types of numbers
– Exact
– Measured
► Exact numbers are infinitely important
► Measured number = they are measured with a measuring
device (name all 4) so these numbers have ERROR.
► When you use your calculator your answer can only be as
accurate as your worst measurement…Doohoo 
Chapter Two
53
Exact Numbers
An exact number is obtained when you count objects
or use a defined relationship.
Counting objects are always exact
2 soccer balls
4 pizzas
Exact relationships, predefined values, not measured
1 foot = 12 inches
1 meter = 100 cm
For instance is 1 foot = 12.000000000001 inches? No
1 ft is EXACTLY 12 inches.
54
Learning Check
A. Exact numbers are obtained by
1. using a measuring tool
2. counting
3. definition
B. Measured numbers are obtained by
1. using a measuring tool
2. counting
3. definition
55
Solution
A. Exact numbers are obtained by
2. counting
3. definition
B. Measured numbers are obtained by
1. using a measuring tool
56
Learning Check
Classify each of the following as an exact or a
measured number.
1 yard = 3 feet
The diameter of a red blood cell is 6 x 10-4 cm.
There are 6 hats on the shelf.
Gold melts at 1064°C.
57
Solution
Classify each of the following as an exact (1) or a
measured(2) number.
This is a defined relationship.
A measuring tool is used to determine length.
The number of hats is obtained by counting.
A measuring tool is required.
58
Measured Numbers
►Do you see why Measured Numbers have error…you
have to make that Guess!
►All but one of the significant figures are known with
certainty. The last significant figure is only the best
possible estimate.
►To indicate the precision of a measurement, the
value recorded should use all the digits known with
certainty.
59
Note the 4 rules
When reading a measured value, all nonzero digits
should be counted as significant. There is a set of
rules for determining if a zero in a measurement is
significant or not.
► RULE 1. Zeros in the middle of a number are like any
other digit; they are always significant. Thus, 94.072
g has five significant figures.
► RULE 2. Zeros at the beginning of a number are not
significant; they act only to locate the decimal point.
Thus, 0.0834 cm has three significant figures, and
0.029 07 mL has four.
Chapter Two
60
► RULE 3. Zeros at the end of a number and after
the decimal point are significant. It is assumed
that these zeros would not be shown unless they
were significant. 138.200 m has six significant
figures. If the value were known to only four
significant figures, we would write 138.2 m.
► RULE 4. Zeros at the end of a number and before
an implied decimal point may or may not be
significant. We cannot tell whether they are part
of the measurement or whether they act only to
locate the unwritten but implied decimal point.
Chapter Two
61
Practice Rule #1 Zeros
45.8736
6
•All digits count
.000239
3
•Leading 0’s don’t
.00023900 5
•Trailing 0’s do
48000.
5
•0’s count in decimal form
48000
2
•0’s don’t count w/o decimal
3.982106 4
1.00040
6
•All digits count
•0’s between digits count as
well as trailing in decimal form
Rounding Off Numbers
► Often when doing arithmetic on a pocket
calculator, the answer is displayed with more
significant figures than are really justified.
► How do you decide how many digits to keep?
► Simple rules exist to tell you how.
Chapter Two
63
► Once you decide how many digits to retain, the
rules for rounding off numbers are straightforward:
► RULE 1. If the first digit you remove is 4 or less, drop
it and all following digits. 2.4271 becomes 2.4 when
rounded off to two significant figures because the
first dropped digit (a 2) is 4 or less.
► RULE 2. If the first digit removed is 5 or greater,
round up by adding 1 to the last digit kept. 4.5832 is
4.6 when rounded off to 2 significant figures since
the first dropped digit (an 8) is 5 or greater.
► If a calculation has several steps, it is best to round
off at the end.
Chapter Two
64
Practice Rule #2 Rounding
Make the following into a 3 Sig Fig number
1.5587
1.56
.0037421
.00374
1367
1370
128,522
129,000
1.6683 106
1.67 106
Your Final number
must be of the same
value as the number
you started with,
129,000 and not 129
Examples of Rounding
For example you want a 4 Sig Fig number
0 is dropped, it is <5
4965.03
4965
780,582
780,600 8 is dropped, it is >5; Note you
must include the 0’s
1999.5
2000.
5 is dropped it is = 5; note you
need a 4 Sig Fig
RULE 1. In carrying out a multiplication or division,
the answer cannot have more significant figures than
either of the original numbers.
Chapter Two
67
►RULE 2. In carrying out an addition or
subtraction, the answer cannot have more digits
after the decimal point than either of the
original numbers.
Chapter Two
68
Multiplication and division
32.27  1.54 = 49.6958
49.7
3.68  .07925 = 46.4353312
46.4
1.750  .0342000 = 0.05985
.05985
3.2650106  4.858 = 1.586137  107
1.586 107
6.0221023  1.66110-24 = 1.000000
1.000
Addition/Subtraction
25.5
+34.270
59.770
59.8
32.72
- 0.0049
32.7151
32.72
320
+ 12.5
332.5
330
Addition and Subtraction
.56
__ + .153
___ = .713
__ .71
82000 + 5.32 = 82005.32
82000
10.0 - 9.8742 = .12580
.1
10 – 9.8742 = .12580
0
Look for the
last important
digit
Mixed Order of Operation
8.52 + 4.1586  18.73 + 153.2 =
= 8.52 + 77.89 + 153.2 = 239.61 =
239.6
(8.52 + 4.1586)  (18.73 + 153.2) =
= 12.68  171.9 = 2179.692 =
2180.
1.3 Measurement
Limits of Measurement
If the least precise measurement in a
calculation has three significant figures,
then the calculated answer can have at
most three significant figures.
• Mass = 34.73 grams
• Volume = 4.42 cubic centimeters.
•
Rounding to three significant figures, the
density is 7.86 grams per cubic centimeter.
1.3 Measurement
Limits of Measurement
Accuracy
Another important quality in a measurement
is its accuracy. Accuracy is the closeness
of a measurement to the actual value of
what is being measured.
For example, suppose a digital clock is
running 15 minutes slow. Although the clock
would remain precise to the nearest second,
the time displayed would not be accurate.
1.3 Measurement
Measuring Temperature
A thermometer is an instrument that
measures temperature, or how hot an object
is.
1.3 Measurement
Measuring Temperature
Celsius (centigrade)
temperature scale
Fahrenheit scale
Capillary tube
Colored liquid The liquid
moves up and down the
capillary tube as the
temperature changes.
Bulb The bulb
contains the
reservoir of liquid.
Scale The scale indicates the
temperature according to how
far up or down the capillary
tube the liquid has moved.
1.3 Measurement
Measuring Temperature
Compressed
scale
Liquid rises
less in a
wide tube
for the same
temperature
change.
Liquid rises
more in a
narrow tube
for the same
temperature
change.
Expanded,
easy-to-read scale
1.3 Measurement
Measuring Temperature
The two temperature scales that you are probably
most familiar with are the Fahrenheit scale and the
Celsius scale.
• A degree Celsius is almost twice as large as a degree
Fahrenheit.
• You can convert from one scale to the other by using one
of the following formulas.
1.3 Measurement
Measuring Temperature
The SI base unit for temperature is the
kelvin (K).
• A temperature of 0 K, or 0 kelvin, refers to the
lowest possible temperature that can be
reached.
• In degrees Celsius, this temperature is
–273.15°C. To convert between kelvins and
degrees Celsius, use the formula:
1.3 Measurement
Measuring Temperature
Temperatures can be expressed in degrees
Fahrenheit, degrees Celsius, or kelvins.
Measurement and Significant Figures
► Every experimental
measurement has a
degree of uncertainty.
► The volume, V, at right
is certain in the 10’s
place, 10mL<V<20mL
► The 1’s digit is also
certain, 17mL<V<18mL
► A best guess is needed
for the tenths place.
Chapter Two
81
What is the Length?
1
•
•
•
•
•
82
2
3
We can see the markings between 1.6-1.7cm
We can’t see the markings between the .6-.7
We must guess between .6 & .7
We record 1.67 cm as our measurement
The last digit an 7 was our guess...stop there
4 cm
Learning Check
What is the length of the wooden stick?
1) 4.5 cm
2) 4.54 cm
3) 4.547 cm
? 8.00 cm or 3 (2.2/8)
84
Let’s Practice
• What is the dimensions of an index card in
inches?
• Convert each dimension to centimeters. The
conversion factor is 1 in. = 2.54 cm.
• Check your answer.
• Measure with various instruments using
accuracy and precision in your measurements.
• Activity Relating to 1.3
1.3 Measurement
Assessment Questions
1. A shopping mall has a length of 200 meters and a
width of 75 meters. What is the area of the mall, in
scientific notation?
a.
b.
c.
d.
1 × 103 m2
1.5 × 103 m2
1.5 × 104 m2
1.75 × 104 m2
1.3 Measurement
Assessment Questions
1. A shopping mall has a length of 200 meters and a
width of 75 meters. What is the area of the mall, in
scientific notation?
a.
b.
c.
d.
1 × 103 m2
1.5 × 103 m2
1.5 × 104 m2
1.75 × 104 m2
ANS: C
1.3 Measurement
Assessment Questions
2. A student measures the volume and mass of a
liquid. The volume is 50.0 mL and the mass is
78.43 g. What is the correct calculated value of
the liquid’s density? (A calculator reads 1.5686.)
a.
b.
c.
d.
1.6 g/cm3
1.57 g/cm3
1.569 g/cm3
1.5686 g/cm3
1.3 Measurement
Assessment Questions
2. A student measures the volume and mass of a
liquid. The volume is 50.0 mL and the mass is
78.43 g. What is the correct calculated value of
the liquid’s density? (A calculator reads 1.5686.)
a.
b.
c.
d.
1.6 g/cm3
1.57 g/cm3
1.569 g/cm3
1.5686 g/cm3
ANS: B
1.3 Measurement
Assessment Questions
3. How can you convert a temperature expressed in
kelvin (K) to degree Celsius (°C)?
a.
b.
c.
d.
add 32
subtract 32
add 273
subtract 273
1.3 Measurement
Assessment Questions
3. How can you convert a temperature expressed in
kelvin (K) to degree Celsius (°C)?
a.
b.
c.
d.
add 32
subtract 32
add 273
subtract 273
ANS: C
1.3 Measurement
Assessment Questions
1. The SI base unit for length is the mile.
True
False
1.3 Measurement
Assessment Questions
1. The SI base unit for length is the mile.
True
False
ANS:
F, meter
1.3 Measurement
Homework
• Section 1.3
Measurement